Classification and properties of the \(\pi \)submaximal subgroups in minimal nonsolvable groups
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Abstract
Let \(\pi \) be a set of primes. According to H. Wielandt, a subgroup H of a finite group X is called a \(\pi \)submaximal subgroup if there is a monomorphism \(\phi :X\rightarrow Y\) into a finite group Y such that \(X^\phi \) is subnormal in Y and \(H^\phi =K\cap X^\phi \) for a \(\pi \)maximal subgroup K of Y. In his talk at the celebrated conference on finite groups in SantaCruz (USA) in 1979, Wielandt posed a series of open questions and among them the following problem: to describe the \(\pi \)submaximal subgroup of the minimal nonsolvable groups and to study properties of such subgroups: the pronormality, the intravariancy, the conjugacy in the automorphism group etc. In the article, for every set \(\pi \) of primes, we obtain a description of the \(\pi \)submaximal subgroup in minimal nonsolvable groups and investigate their properties, so we give a solution of Wielandt’s problem.
Keywords
Minimal nonsolvable group Minimal simple group \(\pi \)Maximal subgroup \(\pi \)Submaximal subgroup Pronormal subgroup1 Introduction
In the paper, all groups are finite and G always denotes a finite group. Moreover, \(\pi \) denotes some given set of primes and \(\pi '\) is the set of primes p such that \(p\notin \pi \). For a natural number n, let \(\pi (n)\) be the set of prime divisors of n and let \(n_\pi \) be the \(\pi \)share of n, that is, the greatest divisor m of n such that \(\pi (m)\subseteq \pi \). Clearly, \(n=n_\pi n_{\pi '}\) and \((n_\pi , n_{\pi '})=1\). For a group G, let \(\pi (G)=\pi (G)\). A group G is called a \(\pi \)group if \(\pi (G)\subseteq \pi \).
A subgroup H of G is said to be \(\pi \)maximal if H is maximal with respect to inclusion in the set of \(\pi \)subgroups of G. We denote by \({{\mathrm{m}}}_\pi (G)\) the set of \(\pi \)maximal subgroups of G.
 (1)
G possesses a \(\pi \)Hall subgroup (that is, a subgroup of order \(G_\pi \)),
 (2)
every two \(\pi \)Hall subgroups are conjugate in G,
 (3)
every \(\pi \)subgroup of G is contained in some \(\pi \)Hall subgroup.
We denote by \({{\mathrm{Hall}}}_\pi (G)\) the set of \(\pi \)Hall subgroups of a group G. The \(\mathscr {D}_\pi \)groups and the groups containing \(\pi \)Hall subgroups are well investigated (see a survey [31] and next works [7, 10, 25, 34]). In particular, it is proved that \(G\in \mathscr {D}_\pi \) if and only if every composition factor of G is a \(\mathscr {D}_\pi \)groups (see [22, Theorem 7.7], [31, Theorem 6.6], [6, Chapter 2, Theorem 6.15]) and the simple \(\mathscr {D}_\pi \)groups are described in [20, Theorem 3] (see also [31, Theorem 6.9]). These results are based on the description of Hall subgroups in the finite simple groups and the following nice properties of \(\pi \)Hall subgroups: if N is normal and H is \(\pi \)Hall subgroups in G, then \(HN/N\in {{\mathrm{Hall}}}_\pi (G/N)\) and \(H\cap N\in {{\mathrm{Hall}}}_\pi (N)\).
The Hall–Chunikhin Theorem shows that, if G is nonsolvable, then \(G\notin \mathscr {D}_\pi \) for some \(\pi \). It is clear that \({{\mathrm{m}}}_\pi (G)\ne \varnothing \) and \({{\mathrm{Hall}}}_\pi (G)\subseteq {{\mathrm{m}}}_\pi (G)\) for any group G but, in general, \({{\mathrm{m}}}_\pi (G)\nsubseteq {{\mathrm{Hall}}}_\pi (G)\) and it may happen that \({{\mathrm{Hall}}}_\pi (G)=\varnothing \). It is natural to try to find a description for \(\pi \)maximal subgroups similar to ones for \(\pi \)Hall subgroups. A hardness is that the \(\pi \)maximal subgroups have no properties similar to the mentioned above properties of \(\pi \)Hall subgroups.
For example [38, (4.2)], if X and Y are groups and \(X\notin \mathscr {D}_\pi \) for some \(\pi \), then, for any \(\pi \)subgroup (not only for \(\pi \)maximal) K of Y, there is a \(\pi \)maximal subgroup H of the regular wreath product \(X\wr Y\) such that the image of H under the natural epimorphism \(X\wr Y\rightarrow Y\) coincides with K.
In general, the intersection of a \(\pi \)maximal subgroup H with a normal subgroup N of G is not a \(\pi \)maximal subgroup of N. For example, it is easy to show that a Sylow 2subgroup H of \(G=PGL_2(7)\) is \(\{2,3\}\)maximal in G but \(H\cap N\notin {{\mathrm{m}}}_{\{2,3\}}(N)\) for \(N=PSL_2(7)\).
But the situation with intersections of \(\pi \)maximal and normal subgroups of a finite group is not so dramatic in comparing with the situation with images under homomorphisms since the following statement holds.
Proposition 1.1
(Wielandt–Hartley Theorem) Let G be a finite group, let N be a subnormal subgroup of G and \(H\in {{\mathrm{m}}}_\pi (G)\). Then \(H\cap N=1\) if and only if N is a \(\pi '\)group.
For the case where N normal in G, Wielandt’s proof of this statement can be found in [39, 13.2], and Hartly’s one in [11, Lemmas 2 and 3]. The proof of the above proposition in the general case see [26, Theorem 7].
In view of Proposition 1.1, it is natural to consider the following concept. According to H. Wielandt, a subgroup H of a group X is called a \(\pi \)submaximal subgroup if there is a monomorphism \(\phi :X\rightarrow Y\) into a group Y such that \(X^\phi \) is subnormal in Y and \(H^\phi =K\cap X^\phi \) for a \(\pi \)maximal subgroup K of Y. We denote by \({{\mathrm{sm}}}_\pi (X)\) the set of \(\pi \)submaximal subgroups of X.
Evidently, \({{\mathrm{m}}}_\pi (G)\subseteq {{\mathrm{sm}}}_\pi (G)\) for any group G. The inverse inclusion does not hold in general as one can see in the above example: any Sylow 2subgroup of \(PSL_2(7)\) is \(\{2,3\}\)submaximal but is not \(\{2,3\}\)maximal.
By using the closeness of the class of \(\mathscr {D}_\pi \)group under taking extensions [22, Theorem 7.7], (see also[31, Theorem 6.6] and [6, Chapter 2, Theorem 6.15]) one can show that \(G\in \mathscr {D}_\pi \) if and only if every two \(\pi \)submaximal subgroup of G are conjugate. In particular, if \(G\in \mathscr {D}_\pi \), then \({{\mathrm{sm}}}_\pi (G)={{\mathrm{m}}}_\pi (G)={{\mathrm{Hall}}}_\pi (G)\).
In view of the Hall–Chunikhin Theorem, it is natural to consider some “critical” situation where G is nonsolvable (and \(G\notin \mathscr {D}_\pi \) for some \(\pi \)) but all subgroups of G are solvable (and so they are \(\mathscr {D}_\pi \)groups). In this situation, G possesses more than one conjugacy class of \(\pi \)submaximal subgroups. In the paper, we consider the following problem which was possed by H. Wielandt in his talk^{2} at the wellknown conference on finite groups in SantaCruz in 1979 [38, Frage (g)]:
Problem 1
(H. Wielandt 1979) To describe the \(\pi \)submaximal subgroups of the minimal nonsolvable groups. To study properties of such subgroups: conjugacy classes, the pronormality, the intravariancy,^{3} the conjugacy in the automorphism group etc.
 (1)
\(L_2(2^p)\) where p is a prime;
 (2)
\(L_2(3^p)\) where p is an odd prime;
 (3)
where p is a prime such that \(p>3\) and \(p^2+1\equiv 0\pmod 5\);
 (4)
\(Sz(2^p)\) where p is an odd prime;
 (5)
\(L_3(3)\).
In the paper, for every set \(\pi \) of primes and for every minimal nonsolvable group G, we solve Problem 1. More precisely, in the first, we reduce this problem to the case where G is a minimal simple group by proving the following statement:
Proposition 1.2
 (1)
\({{\mathrm{m}}}_\pi (G/N)=\{KN/N\mid K\in {{\mathrm{m}}}_\pi (G)\}\).
 (2)
\({H\cap N}\) coincides with the \(\pi \)Hall subgroup \(O_\pi (N)\) of N.
 (3)
\(HN/N\in {{\mathrm{sm}}}_\pi (G/N)\).
 (4)
HN / N is pronormal in G / N if and only if H is pronormal in G.
 (5)
H is intravariant in G if and only if the conjugacy class of HN / N in G / N is invariant under the image \({\overline{{{\mathrm{Aut}}}(G)}}\) of the map \({{\mathrm{Aut}}}(G)\rightarrow {{\mathrm{Aut}}}(G/N)\) given by the rule \(\phi \mapsto \bar{\phi }\) where \(\bar{\phi }:Ng\mapsto Ng^\phi \) for \(\phi {\in {{\mathrm{Aut}}}(G)}\).
If G is a minimal nonsolvable group, then \(F(G)=\Phi (G)\). Thus, for every \(\pi \)submaximal subgroups H of a minimal nonsolvable group G, the image of H in the corresponding minimal simple group \(\tilde{G}=G/\Phi (G)\) is a \(\pi \)submaximal subgroup of \(\tilde{G}\). In order to solve Problem 1, we only need to describe the \(\pi \)submaximal subgroup of the minimal simple groups, that is, in the groups of the Thompson list. It is known [5, Corollary 1.7.10] that \(\pi (G)=\pi (\tilde{G})\), and it is clear that, if \(\pi \cap \pi (\tilde{G})=\varnothing \), then \({{\mathrm{sm}}}_\pi (G)={{\mathrm{m}}}_\pi (G)=\{1\}\); if \(\pi \cap \pi (\tilde{G})=\{p\}\), then \({{\mathrm{sm}}}_\pi (G)={{\mathrm{m}}}_\pi (G)={{\mathrm{Syl}}}_p(G)\); and if \(\pi (\tilde{G})\subseteq \pi \), then \({{\mathrm{sm}}}_\pi (G)={{\mathrm{m}}}_\pi (G)=\{G\}\). A description of the \(\pi \)submaximal subgroup in the minimal simple groups for the remaining cases is given in the following Theorem.
Theorem 1.1
Let \(\pi \) be a set of primes and S a minimal simple group. Suppose that \({\pi \cap \pi (S)}>1\) and \(\pi (S)\nsubseteq \pi \). Then representatives of the conjugacy classes of \(\pi \)submaximal subgroups of S, the information of their structure, \(\pi \)maximality, pronormality, intravariancy, and the action of \({{\mathrm{Aut}}}(S)\) on the set of conjugacy classes of \(\pi \)submaximal subgroups can be specify in the corresponding Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11 below.
Corollary 1.1
For every set \(\pi \) of primes, the \(\pi \)submaximal subgroups of any minimal nonsolvable group are pronormal.
Note that our results depend on Thompson’s classification of the minimal simple groups and do not depend on the classification of finite simple groups.
1.1 Notation in Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11

\(\varepsilon \) denotes either \(+1\) or \(1\) and the sing of this number.

\(C_n\) denotes the cyclic group of order n.

\(E_q\) denotes the elementary abelian group of order q where q is a power of a prime.

\(D_{2n}\) denotes the dihedral group of order 2n, i. e. \(D_{2n}=\langle x,y\mid x^n=y^2=1, x^y= zx^{1}\rangle \). Note that \(D_4\cong E_4\).

\(SD_{2^n}\) denotes the semidihedral group of order \(2^n\), i. e.
\(SD_{2^n}=\langle x,y\mid x^{2^{n1}}=y^2=1, x^y=x^{2^{n2}+1}\rangle \).

\(S_n\) means a symmetric group of degree n.

\(A_n\) denotes the alternating group of degree n.

\(GL_n(q)\) denotes the general linear group of degree n over the field of order q.

\(PGL_n(q)\) denotes the projective general linear group of degree n over the field of order q.

\(L_n(q)=PSL_n(q)\) denotes the projective special linear group of degree n over the field of order q.

\(r_+^{1+2n}\) denotes the extra special group of order \(r^{1+2n}\) and of exponent r where r is an odd prime.

A : B means a split extension of a group A by a group B (A is normal).

\(A^n\) denotes the direct product of n copies of A.

\(A^{m+n}\) means \(A^m:A^{n}\).

The conditions in the column “Cond.” are necessary and sufficient for the existence and the \(\pi \)submaximality of corresponding H. If a cell in this column is empty, then it means that the corresponding \(\pi \)submaximal subgroup always exists.

In the column “H” the structure of corresponding H is given.

The conditions in the column “is not \(\pi \)max. if” are necessary and sufficient for corresponding H to be not \(\pi \)maximal in S. If either this column is skipped or a cell in this column is empty, then the corresponding subgroup is \(\pi \)maximal.

A number n in the column “NCC” is equal to the number of conjugacy classes of \(\pi \)submaximal subgroups of S isomorphic to corresponding subgroup H and, if \(n>1\), then in the same column the action of \({{\mathrm{Aut}}}(S)\) on these classes is described.

The symbol “\(\checkmark \)” in the column “Pro.” means that the corresponding subgroup H is pronormal in S.

The symbol “\(\checkmark \)” in the column “Intra.” means that the corresponding subgroup H is intravariant in S. If a cell in this column is empty, then H is not intravariant.
1.2 The \(\pi \)submaximal subgroups in \(S=L_2(q)\), where \(q=2^p\), p is prime, for \(\pi \) such that \(\pi \cap \pi (S)>1\) and \(\pi (S)\nsubseteq \pi \)
The \(\pi \)submaximal subgroups of \(S=L_2(q)\), where \(q=2^p\), p is a prime. Case: \(2\notin \pi \). Notation: \(\pi _\varepsilon =\pi \cap \pi (q\varepsilon )\), \(\varepsilon \in \{+,\}\). In any cases H is \(\pi \)maximal
Cond.  H  NCC  Pro.  Intra.  

1  \(\pi _+\ne \varnothing \)  \(C_{(q1)_\pi }\)  1  \(\checkmark \)  \(\checkmark \) 
2  \(\pi _\ne \varnothing \)  \(C_{(q+1)_\pi }\)  1  \(\checkmark \)  \(\checkmark \) 
The \(\pi \)submaximal subgroups of \(S=L_2(q)\), where \(q=2^p\), p is a prime. Case: \(2\in \pi \). Notation: \(\pi _\varepsilon =\pi \cap \pi (q\varepsilon )\), \(\varepsilon \in \{+,\}\). In any cases H is \(\pi \)maximal
Cond.  H  NCC  Pro.  Intra.  

1  \(E_q:C_{(q1)_\pi }\)  1  \(\checkmark \)  \(\checkmark \)  
2  \(\pi _+\ne \varnothing \)  \(D_{2(q1)_\pi }\)  1  \(\checkmark \)  \(\checkmark \) 
3  \(\pi _\ne \varnothing \)  \(D_{2(q+1)_\pi }\)  1  \(\checkmark \)  \(\checkmark \) 
The \(\pi \)submaximal subgroups of \(S=L_2(q)\), where \(q=3^p\), p is an odd prime. Case: \(2\notin \pi \). Notation: \(\pi _\varepsilon =\pi \cap \pi (q\varepsilon )\), \(\varepsilon \in \{+,\}\). In any cases H is \(\pi \)maximal
Cond.  H  NCC  Pro.  Intra.  

1  \(3\in \pi \)  \(E_q:C_{(q1)_\pi }\)  1  \(\checkmark \)  \(\checkmark \) 
2  \(3\notin \pi \) and \(\pi _+\ne \varnothing \)  \(C_{(q1)_\pi }\)  1  \(\checkmark \)  \(\checkmark \) 
3  \(\pi _\ne \varnothing \)  \(C_{(q+1)_\pi }\)  1  \(\checkmark \)  \(\checkmark \) 
The \(\pi \)submaximal subgroups of \(S=L_2(q)\), where \(q=3^p\), p is an odd prime. Case: \(2\in \pi \). Notation: \(\pi _\varepsilon =\pi \cap \pi (q\varepsilon )\), \(\varepsilon \in \{+,\}\) \(\phi \in {{\mathrm{Aut}}}(S){\setminus }{{\mathrm{Inn}}}(S)\), \(\phi =p\). In any cases H is \(\pi \)maximal
Cond.  H  NCC  Pro.  Intra.  

1  \(3\in \pi \)  \(E_q:C_{\frac{1}{2}(q1)_\pi }\)  1  \(\checkmark \)  \(\checkmark \) 
2  \(\pi _+\ne \{2\}\)  \(D_{(q1)_\pi }\)  1  \(\checkmark \)  \(\checkmark \) 
3  \(D_{(q+1)_\pi }\)  1  \(\checkmark \)  \(\checkmark \)  
4  \(3\in \pi \)  \(A_4\)  1  \(\checkmark \)  \(\checkmark \) 
1.3 The \(\pi \)submaximal subgroups in \(S=L_2(q)\), where \(q=3^p\), p is odd prime, for \(\pi \) such that \(\pi \cap \pi (S)>1\) and \(\pi (S)\nsubseteq \pi \)
The \(\pi \)submaximal subgroups of \(S=L_2(q)\), where \(q>3\) is a prime, \(q^2\equiv 1\pmod 5\). Case: \(2\notin \pi \). Notation: \(\pi _\varepsilon =\pi \cap \pi (q\varepsilon )\), \(\varepsilon \in \{+,\}\). In any cases H is \(\pi \)maximal
Cond.  H  NCC  Pro.  Intra.  

1  \(q\in \pi \)  \(C_q:C_{(q1)_\pi }\)  1  \(\checkmark \)  \(\checkmark \) 
2  \(q\notin \pi \) and \(\pi _+\ne \varnothing \)  \(C_{(q1)_\pi }\)  1  \(\checkmark \)  \(\checkmark \) 
3  \(\pi _\ne \varnothing \)  \(C_{(q+1)_\pi }\)  1  \(\checkmark \)  \(\checkmark \) 
1.4 The \(\pi \)submaximal subgroups in \(S=L_2(q)\), where q is a prime, \(q^2\equiv 1\pmod 5\), for \(\pi \) such that \(\pi \cap \pi (S)>1\) and \(\pi (S)\nsubseteq \pi \)
The \(\pi \)submaximal subgroups of \(S=L_2(q)\), where \(q>3\) is a prime, \(q^2\equiv 1\pmod 5\), in the case \(2\in \pi \). Notation: \(\pi _\varepsilon =\pi \cap \pi (q\varepsilon )\), \(\varepsilon \in \{+,\}\), \(\delta \in {{\mathrm{Aut}}}(S){\setminus }{{\mathrm{Inn}}}(S)\), \(\delta =2\), \({{\mathrm{Aut}}}(S)=\langle {{\mathrm{Inn}}}(S),\delta \rangle \cong S:\langle \delta \rangle \cong PGL_2(q)\)
Cond.  H  NCC  H is not \(\pi \)max. if  Pro.  Intra.  

1  \(q\in \pi \)  \(C_q:C_{\frac{1}{2}(q1)_\pi }\)  1  \(\checkmark \)  \(\checkmark \)  
2  either \(\pi _+\ne \{2\}\), or \(3\notin \pi \) or \(q\equiv 1\pmod 8\)  \(D_{(q1)_\pi }\)  1  either \(\pi _+=\{2\}, 3\in \pi \), and \(q\equiv 41\pmod {48}{}^*\) or \(\pi _+=\{2,3\}\) and \(q\equiv 7,31\pmod {72}{}^{**}\)  \(\checkmark \)  \(\checkmark \) 
3  either \(\pi _\ne \{2\},\) or \(3\notin \pi \), or \(q\equiv 1\pmod 8\)  \(D_{(q+1)_\pi }\)  1  either \(\pi _=\{2\}\), \(3\in \pi \), and \(q\equiv 7\pmod {48}{}^*\) or \(\pi _=\{2,3\}\) and \(q\equiv 41,65\pmod {72}{}^{**}\)  \(\checkmark \)  \(\checkmark \) 
4  \(3\in \pi \) and \(q\equiv \pm 3\pmod 8\)  \(A_4\)  1  \(\checkmark \)  \(\checkmark \)  
5  \(3\in \pi \) and \(q\equiv \pm 1\pmod 8\)  \(S_4\)  2 permuted by \(\delta \)  \(\checkmark \) 
1.5 The submaximal \(\pi \)subgroups in \(S=Sz(q)\), where \(q=2^p\), p is odd prime, for \(\pi \) such that \(\pi \cap \pi (S)>1\) and \(\pi (S)\nsubseteq \pi \)
The \(\pi \)submaximal subgroups of \(S=Sz(q)\), where \(q=2^p\), p is an odd prime. Case: \(2\notin \pi \). Notation: \(r=\sqrt{(2q)}=2^{(p+1)/2}\) \(\pi _0=\pi \cap \pi (q1)\), \(\pi _\varepsilon =\pi \cap \pi (q\varepsilon r+1)\), \(\varepsilon \in \{+,\}\). In any cases H is \(\pi \)maximal
Cond.  H  NCC  Pro.  Intra.  

1  \(\pi _0\ne \varnothing \)  \(C_{(q 1)_\pi }\)  1  \(\checkmark \)  \(\checkmark \) 
2  \(\pi _+\ne \varnothing \)  \(C_{(q r+1)_\pi }\)  1  \(\checkmark \)  \(\checkmark \) 
3  \(\pi _\ne \varnothing \)  \(C_{(q r+1)_\pi }\)  1  \(\checkmark \)  \(\checkmark \) 
The \(\pi \)submaximal subgroups of \(S=Sz(q)\), where \(q=2^p\), p is an odd prime. Case: \(2\in \pi \). Notation: \(r=\sqrt{(2q)}=2^{(p+1)/2}\) \(\pi _0=\pi \cap \pi (q1)\), \(\pi _\varepsilon =\pi \cap \pi (q\varepsilon r+1)\), \(\varepsilon \in \{+,\}\). In any cases H is \(\pi \)maximal
Cond.  H  NCC  Pro.  Intra.  

1  \(E_q^{1+1}:C_{(q1)_\pi }\)  1  \(\checkmark \)  \(\checkmark \)  
2  \(\pi _0\ne \varnothing \)  \(C_{(q1)_\pi }:C_4\)  1  \(\checkmark \)  \(\checkmark \) 
3  \(\pi _+\ne \varnothing \)  \(D_{2(q r+1)_\pi }\)  1  \(\checkmark \)  \(\checkmark \) 
4  \(\pi _\ne \varnothing \)  \(D_{2(q+ r+1)_\pi }\)  1  \(\checkmark \)  \(\checkmark \) 
1.6 The \(\pi \)submaximal subgroups of \(S=L_3(3)\), for \(\pi \) such that \(\pi \cap \pi (S)>1\) and \(\pi (S)\nsubseteq \pi \)
The \(\pi \)submaximal subgroups of \(S=L_3(3)\). Case: \(\pi \cap \pi (S)=\{3,13\}\). In any cases H is \(\pi \)maximal
H  NCC  Pro.  Intra.  

1  \(C_{13}:C_{3}\)  1  \(\checkmark \)  \(\checkmark \) 
2  \(3_+^{2+1}\)  1  \(\checkmark \)  \(\checkmark \) 
The \(\pi \)submaximal subgroups of \(S=L_3(3)\). Case: \(\pi \cap \pi (S)=\{2,13\}\). In any cases H is \(\pi \)maximal
H  NCC  Pro.  Intra.  

1  \(C_{13}\)  1  \(\checkmark \)  \(\checkmark \) 
2  \({SD}_{16}\)  1  \(\checkmark \)  \(\checkmark \) 
The \(\pi \)submaximal subgroups of \(S=L_3(3)\). Case: \(\pi \cap \pi (S)=\{2,3\}\) \(\gamma \in {{\mathrm{Aut}}}(S){\setminus }{{\mathrm{Inn}}}(S)\), \(\gamma =2\), \({{\mathrm{Aut}}}(S)=\langle {{\mathrm{Inn}}}(S),\gamma \rangle \)
H  NCC  H is not \(\pi \)max. if  Pro.  Intra.  

1  \(E_{3^2}:GL_2(3)\)  2 permuted by \(\gamma \)  \(\checkmark \)  
2  \(3_+^{1+2}:C_{2}^2\)  1  always; \(H\le E_{3^2}:GL_2(3)\)  \(\checkmark \)  \(\checkmark \) 
3  \(GL_2(3)\)  1  always; \(H\le E_{3^2}:GL_2(3)\)  \(\checkmark \)  \(\checkmark \) 
4  \(S_4\)  1  \(\checkmark \)  \(\checkmark \) 
2 Preliminaries
We write \(M\lessdot G\) if M is a maximal subgroup of G, that is, \(M<G\) and \(M \le H \le G\) implies that either \(H=M\) or \(H=G\). Moreover, we write \(H\unlhd G\) and \(H{\unlhd \unlhd }G\) if H is a normal or subnormal subgroup of G, respectively.
Lemma 2.1
Let S be a minimal simple group. Then representatives of the conjugacy classes of maximal subgroups of S, the information on their structure, conjugacy classes, and the action of \({{\mathrm{Aut}}}(S)\) on the set of the conjugacy classes of maximal subgroups can be specify in the corresponding Tables 12, 13, 14, 15 and 16.
Proof
See [1, Theorem 2.1.1, Tables 8.1–8.4 and 8.16], [16, Theorem II.8.27], [2, 27, Theorem 9]. \(\square \)
Maximal subgroups of \(S=L_2(q)\) where \(q=2^p\), p is prime
M  NCC 

\(E_q:C_{(q1)}\)  1 
\(D_{2(q1)}\)  1 
\(D_{2(q+1)}\)  1 
Maximal subgroups of \(S=L_2(q)\) where \(q=3^p\), p is an odd prime
M  NCC 

\(C_q:C_{\frac{1}{2}(q1)}\)  1 
\(D_{q1}\)  1 
\(D_{q+1}\)  1 
\(A_4\)  1 
Maximal subgroups of \(S=L_2(q)\) where \(q>3\) is prime, \(q^2\equiv 1\pmod 5\)
M  NCC  Conditions 

\(C_q:C_{\frac{1}{2}(q1)}\)  1  
\(D_{(q1)}\)  1  \(q\ne 7\) 
\(D_{(q+1)}\)  1  \(q\ne 7\) 
\(A_4\)  1  \(q\equiv \pm 3\pmod 8\) 
\(S_4\)  2 permuted by \(\delta \)  \(q\equiv \pm 1\pmod 8\) 
Maximal subgroups of \(S=Sz(q)\) where \(q=2^p\), p is an odd prime
M  NCC 

\(E_q^{1+1}:C_{(q1)}\)  1 
\(C_{(q1)}:C_4\)  1 
\(D_{2(q r+1)}\)  1 
\(D_{2(q+ r+1)}\)  1 
Maximal subgroups of \(S=L_3(3)\)
M  NCC 

\(E_{3^2}:GL_2(3)\)  2 permuted by \(\gamma \) 
\(C_{13}:C_{3}\)  1 
\(S_4\)  1 
Lemma 2.2
 (1)
\(M\cap {{\mathrm{Inn}}}(S)\lessdot {{\mathrm{Inn}}}(S)\);
 (2)
\(S\cong L_2(7)\), \(G={{\mathrm{Aut}}}(S)\), \(M\cong D_{12}\) and \(M\cap {{\mathrm{Inn}}}(S)\cong D_6\);
 (3)
\(S\cong L_2(7)\), \(G={{\mathrm{Aut}}}(S)\), \(M\cong D_{16}\) and \(M\cap {{\mathrm{Inn}}}(S)\cong D_8\);
 (4)
\(S\cong L_3(3)\), \(G={{\mathrm{Aut}}}(S)\), \(M\cong GL_2(3):C_2\) and \(M\cap {{\mathrm{Inn}}}(S)\cong GL_2(3)\);
 (5)
\(S\cong L_3(3)\), \(G={{\mathrm{Aut}}}(S)\), \(M\cong 3_+^{1+2}:D_8\) and \(M\cap {{\mathrm{Inn}}}(S)\cong 3_+^{1+2}:C_2^2\).
Lemma 2.3
(see [35, Statements 7 and 9]) Let \(H_1,\dots ,H_n\) be subnormal subgroups of G. Then \(K=\langle H_1,\dots ,H_n\rangle \) is subnormal in G and every composition factor of K is isomorphic to a composition factor of one of \(H_1,\dots ,H_n\).
Lemma 2.4
(see [15, Lemma 1] and [5, Lemma 1.7.5]) Let A be a a normal subgroup and H be a \(\pi \)Hall subgroup of G. Then \({H\cap A\in {{\mathrm{Hall}}}_\pi (A)}\) and \(HA/A\in {{\mathrm{Hall}}}_\pi (G/A).\)
Lemma 2.5
(see [36] and [5, Theorem 1.10.1]) If G has a nilpotent \(\pi \)Hall subgroup, then \(G\in \mathscr {D}_\pi \).
Lemma 2.6
If G is \(\pi \)separable, then \(G\in \mathscr {D}_\pi \) and \({{\mathrm{sm}}}_\pi (G)={{\mathrm{m}}}_\pi (G)={{\mathrm{Hall}}}_\pi (G)\).
Proof
Lemma 2.7
(see [11, Lemma 2]) Let A be a normal subgroup of G let \(H\in {{\mathrm{m}}}_\pi (G)\). Then \(N_A(H\cap A)/(H\cap A)\) is a \(\pi '\)group.
The following lemma was stated without a proof in [38, 5.3]. Here we give a proof for it.
Lemma 2.8
 (1)
\(H\in {{\mathrm{sm}}}_\pi (S)\).
 (2)
There exists a group G such that S is the socle of G, G / S is a \(\pi \)group, and \(H=S\cap K\) for some \(K\in {{\mathrm{m}}}_\pi (G)\).
 (3)
\(H=S\cap K\) for some \(K\in {{\mathrm{m}}}_\pi ({{\mathrm{Aut}}}(S))\) where S is identified with \({{\mathrm{Inn}}}(S)\).
Proof
\((3)\Rightarrow (2):\) It is sufficient to take \(G=SK\) where K as in (3).
\((2)\Rightarrow (1):\) It follows from the definition.
\((1)\Rightarrow (3)\). Take a group X of the smallest order among all groups G such that S can be embedded into G as a subnormal subgroup and \(H=S\cap K\) for some \(K\in {{\mathrm{m}}}_\pi (G)\). Let \(A=\langle S^X\rangle \). Then the simplicity of S implies that A is a miniml normal subgroup in X. Since X has the smallest possible order, we obtain \(X=KA\).
Thus, we can consider that \(X\le {{\mathrm{Aut}}}(S)\). Let \(M\in {{\mathrm{m}}}_\pi ({{\mathrm{Aut}}}(S))\) such that \(K\le M\). Then \(M\cap S\le S\le X\) and K normalizes \(M\cap S\). This implies that \(M\cap S\le K\) and \(M\cap S=K\cap S =H\). \(\square \)
Lemma 2.9
Let S be a finite simple group and let \(H\in {{\mathrm{sm}}}_\pi (S)\). Then \(N_S(H)/H\) is a \(\pi '\)group.
Proof
We identify S with \({{\mathrm{Inn}}}(S)\). By Lemma 2.8, \(H=K\cap S\), where \(K\in {{\mathrm{m}}}_\pi ({{\mathrm{Aut}}}(S))\). Since \(S\unlhd {{\mathrm{Aut}}}(S)\), Lemma 2.7 implies that \(N_S(H)/H\) is a \(\pi '\)group. \(\square \)
Lemma 2.10
 (1)
\(S\in \mathscr {D}_\pi \).
 (2)
\({{\mathrm{Aut}}}(S)\in \mathscr {D}_\pi \).
 (3)
\(\pi \cap \pi (S)\subseteq \{p\}\cup \pi (q1)\).
Proof
See [21, Theorems A, 2.5, and, 3.3]. \(\square \)
3 Proof of Proposition 1.2
 (1)
\({{\mathrm{m}}}_\pi (G/N)=\{KN/N\mid K\in {{\mathrm{m}}}_\pi (G)\}\).
Proof
 (2)
\({H\cap N}\) coincides with the \(\pi \)Hall subgroup \(O_\pi (N)\) of N.
Proof
Since N is nilpotent, N has a unique \(\pi \)Hall subgroup. Hence we only need to show that \({{H\cap N}\in {{\mathrm{Hall}}}_\pi (N)}\). By property \((*)\) of \(\pi \)submaximal subgroups, \(H\cap N\in {{\mathrm{sm}}}_\pi (N)\) and \({{\mathrm{sm}}}_\pi (N)={{\mathrm{Hall}}}_\pi (N)\) by Lemma 2.6.\(\square \)
 (3)
\(HN/N\in {{\mathrm{sm}}}_\pi (G/N)\).
Proof
In order to prove next statements (4) and (5), we need the following lemma.
Lemma 3.1
In the above notation, HN is \(\pi \)separable and \(H\in {{\mathrm{Hall}}}_\pi (HN)\).
Proof
 (4)
HN / N is pronormal in G / N if and only if H is pronormal in G.
Proof
If H is pronormal in G, then the pronormality of HN / N in G / N is evident.
Conversely, assume that HN / N is pronormal in G / N. Let \(g\in G\). We need to prove that H and \(H^g\) are conjugate in \(\langle H, H^g\rangle \).
Firstly, consider the case when \(g\in N_G(HN)\). Then \(H^g\le HN\) and \(H^g\in {{\mathrm{Hall}}}_\pi (HN)\) by Lemma 3.1. Clearly, the subgroup \(\langle H,H^g\rangle \) of the \(\pi \)separable group HN is also \(\pi \)separable. Lemma 2.6 implies that the \(\pi \)Hall subgroups H and \(H^g\) of \(\langle H,H^g\rangle \) are conjugate in \(\langle H,H^g\rangle \).
Now consider the general case for \(g\in G\). Since HN / N is pronormal in G / N, there is \(y\in \langle H,H^g\rangle \) such that \((HN)^y=(HN)^g\). Hence \(gy^{1}\in N_G(HN)\) and, in view of above, there exist some \(z\in \langle H,H^{gy^{1}}\rangle \le \langle H,H^g\rangle \) such that \(H^z=H^{gy^{1}}\). Hence H and \(H^g\) are conjugate by \(x=zy\in \langle H,H^g\rangle \). \(\square \)
 (5)
H is intravariant in G if and only if the conjugacy class of HN / N in G / N is invariant under \({\overline{{{\mathrm{Aut}}}(G)}}\).
Proof
Recall that \({\overline{{{\mathrm{Aut}}}(G)}}\) is the image in \({{\mathrm{Aut}}}(G/N)\) of \({{\mathrm{Aut}}}(G)\) under the map \(\phi \mapsto \bar{\phi }\) where \(\bar{\phi }:Ng\mapsto Ng^\phi \) for \(\phi {\in {{\mathrm{Aut}}}}(G)\).
Conversely, assume that the conjugacy class of HN / N in G / N is invariant under \({\overline{{{\mathrm{Aut}}}(G)}}\). We need to show that for every \(\phi \in {{\mathrm{Aut}}}(G)\) there exists \(g\in G\) such that \(H^\phi =H^g\). It is clear that \(H^\phi \in {{\mathrm{sm}}}_\pi (G)\). By the hypothesis, there is \(x\in G\) such that \(H^\phi N=H^x N\), and in view of Lemma 3.1, \(H^\phi ,H^x \in {{\mathrm{Hall}}}_\pi (H^\phi N)\) and \(H^\phi N\) is \(\pi \)separable. Hence, \(H^\phi =H^{xy}\) for some \(y\in H^\phi N\) by Lemma 2.6. \(\square \)
4 Proof of Theorem 1.1 and Corollary 1.1
We divide our proof of Theorem 1.1 onto three parts. In the first part (Proposition 4.1 in Sect. 1.1), we prove that if H is a \(\pi \)submaximal subgroup of a minimal simple group S, then H can be found in that of Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11 which corresponds to given S and \(\pi \). In the second part (Proposition 4.2 in Sect. 1.2), we prove that every H in Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11 are \(\pi \)submaximal for corresponding S and prove that the information on the \(\pi \)maximality, conjugacy, intravariancy and the action of \({{\mathrm{Aut}}}(S)\) on the conjugacy classes for this subgroups is true. Finally, in the third part (Proposition 4.3 in Sect. 1.3) we prove the pronormality of the \(\pi \)submaximal subgroups in the minimal simple groups and, as a consequence, in the minimal nonsolvable groups (Corollary 1.1).
4.1 The classification of \(\pi \)submaximal subgroups in minimal simple groups
 (1)
\(L_2(q)\) where \(q=2^p\), p is a prime;
 (2)
\(L_2(q)\) where where \(q=3^p\), p is an odd prime;
 (3)
\(L_2(q)\) where q is a prime such that \(q>3\) and \(q^2+1\equiv 0\pmod 5\);
 (4)
Sz(q) where \(q=2^p\), p is an odd prime;
 (5)
\(L_3(3)\).
Proposition 4.1
If \(H\in {{\mathrm{sm}}}_\pi (S)\) where \(S\in \mathcal T\), then H appears in the corresponding column in that of Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11 which corresponds to S and \(\pi \).
Proof
 (I)
\(M\cap S\lessdot S\);
 (II)
\(S\cong L_2(7)\), \(G={{\mathrm{Aut}}}(S)\), \(M\cong D_{12}\) and \(M\cap S\cong D_6\);
 (III)
\(S\cong L_2(7)\), \(G={{\mathrm{Aut}}}(S)\), \(M\cong D_{16}\) and \(M\cap S\cong D_8\);
 (IV)
\(S\cong L_3(3)\), \(G={{\mathrm{Aut}}}(S)\), \(M\cong GL_2(3):2\) and \(M\cap S\cong GL_2(3)\);
 (V)
\(S\cong L_3(3)\), \(G={{\mathrm{Aut}}}(S)\), \(M\cong 3_+^{1+2}:D_8\) and \(M\cap S\cong 3_+^{1+2}:C_2^2\).
Assume that Case (II) holds. Then \(S=L_2(q)\), where \(q=7\) and \(M\cap S\cong D_6\).
If \(2\notin \pi \), then \(H=3\) and H appears in the 2nd row of Table 5 for \(\pi _+=\{3\}\).
If \(3\notin \pi \), then \(H=2\) and \(2\in \pi \). In this case, H is contained in a Sylow 2subgroup P of S. But \(1<N_P(H)/H \le N_S(H)/H\), so \(N_S(H)/H\) is not a \(\pi '\)group, which contradicts Lemma 2.9. Thus, this case is impossible.
If \(2,3\in \pi \), then \(H=M\cap S\) and H appears in the 2nd row of Table 6 for \(\pi _+\ne \{2\}\).
Assume that Case (III) holds. Then \(S=L_2(q)\), where \(q=7\) and \(M\cap S\cong D_8\). In this case, \(H\in {{\mathrm{Hall}}}_\pi (M\cap S)\) implies that \(2\in \pi \), \(H=M\cap S\) is a Sylow 2subgroup of S, and H appears in the 3rd row of Table 6 for \(q\equiv 1\pmod 4\).
Assume that Case (IV) holds. Then \(S=L_3(3)\) and \(M\cap S\cong GL_2(3)\).
If \(2\notin \pi \), then \(H=3\) and H is contained in a Sylow 3subgroup P of S. But \(1<N_P(H)/H\le N_S(H)/H\), so \(N_S(H)/H\) is not a \(\pi '\)group, which contradicts Lemma 2.9. Thus, this case is impossible.
If \(3\notin \pi \), then \(H\cong SD_{16}\), H coincides with a Sylow 2subgroup of S, and \(\pi \cap \pi (S)=\{2,13\}\). Hence, H appears in the 2nd row of Table 10.
If \(2,3\in \pi \), then \(H=M\cap S\) and H appears in the 3rd row of Table 11 for \(\pi \cap \pi (S)=\{2,3\}\).
Assume that Case (V) holds. Then \(S=L_3(3)\) and \(M\cap S\cong 3_+^{1+2}:C_2^2\).
If \(2\notin \pi \), then \(\pi \cap \pi (S)=\{3,13\}\), \(H\cong 3_+^{1+2}\) and H coincides with a Sylow 3subgroup of S. In this case, H appears in the 2nd row of Table 9.
If \(3\notin \pi \), then \(H=4\) and \(2\in \pi \). In this case, H is contained in a Sylow 2subgroup P of S. But \(1<N_P(H)/H\le N_S(H)/H\). Hence \(N_S(H)/H\) is not a \(\pi '\)group, contrary to Lemma 2.9. Thus, this case is impossible.
If \(2,3\in \pi \), then \(H=M\cap S\) and H appears in the 2nd row of Table 11 for \(\pi \cap \pi (S)=\{2,3\}\).
Now we consider Case (I). In this case, H coincides with some nontrivial \(\pi \)Hall subgroup of a maximal subgroup \(U=M\cap S\) of \(S\in \mathcal T\). By using Lemma 2.1, we consider the nontrivial \(\pi \)Hall subgroups of maximal subgroups U of all groups \(S\in \mathcal T\).
Assume that \(\varepsilon =+\) and H is cyclic of order \((q1)_\pi \). We claim that \(H\notin {{\mathrm{sm}}}_\pi (G)\). Indeed, if \(H\in {{\mathrm{sm}}}_\pi (G)\), then \(H=K\cap S\) for some \(K\in {{\mathrm{m}}}_\pi ({{\mathrm{Aut}}}(S))\) by Lemma 2.8. It is known that \({{{\mathrm{Aut}}}(S)/S=2p}\) (see [1, Table 8.1], for example). Consider two cases: \(p\notin \pi \) and \(p\in \pi \).
If \(p\notin \pi \), then \(K\le S\) and \(H=K\). In particular, in this case, \(H\in {{\mathrm{m}}}_\pi (S)\). But it is easy to see that \(H\in {{\mathrm{Hall}}}_{\pi _+}(S)\) and \(S\in \mathscr {D}_{\pi _+}\) by Lemma 2.5. A maximal subgroup of the type \(E_q:C_{\frac{1}{2}(q1)}\) of S contains some \(\pi _+\)Hall subgroup of S and H is conjugate to this subgroup. It means that H normalizes a Sylow 3subgroup of S, contrary to \(H\in {{\mathrm{m}}}_\pi (S)\).
Thus, if \(2\notin \pi \) and \(3\in \pi ,\) then H is not a cyclic group of order \((q1)_\pi \). In the remaining cases, H appears in Table 3.
Suppose that \(2\in \pi \). Then the maximal subgroups containing a Sylow 2subgroup of S are \(D_{q+1}\) and \(A_4\). Hence H is a 2group if and only if \(\pi _=\{2\}\) and \(H=D_{(q+1)_2}=D_{(q+1)_\pi }\). In particular, such H can not be contained in \(E_q:C_{\frac{1}{2}(q1)}\) and \(D_{q1}\). Now it is easy to see that if \(3\notin \pi \), then H coincides with a \(\pi \)Hall subgroup of \(D_{q\varepsilon }\), where \(\varepsilon \in \{+,\}\); and if \(3\in \pi \), then H can not coincide with a Sylow 2subgroup P by Lemma 2.9 in view of \(N_S(P)\cong A_4\). Hence H must coincides with one of the following groups \(E_q:C_{\frac{1}{2}(q1)_\pi }\), \(D_{(q1)_\pi }\), \(D_{(q+1)_\pi }\) or \(A_4\). In all these cases, H appears in Table 4.
In this case, \({{\mathrm{Aut}}}(S)\cong PGL_2(q)\). By Lemma 2.1, U is one of the following groups: or \(C_q:C_{\frac{1}{2}(q1)}\), or \(D_{q\varepsilon }\) where \(\varepsilon \in \{+,\}\), or one of \(A_4\) and \(S_4\) for \(q\equiv \pm 3\pmod 8\) and \(q\equiv \pm 1\pmod 8\), respectively. It follows from [1, Table 8.1] that, with the exception of the case \(U\cong S_4\), we have \(U=V\cap S\) for some \(V\lessdot {{\mathrm{Aut}}}(S)\) such that \({{\mathrm{Aut}}}(S)=SV\) and \(V:U=2\).
Suppose that \(2\notin \pi \). Since \({{\mathrm{Aut}}}(S):S=2\), H must be a \(\pi \)maximal subgroup of S by Lemma 2.8. If H is contained in one of \(A_4\) or \(S_4\), then \(H=3\) and, since any Sylow 3subgroup of S is cyclic, H is contained in \(D_{q\varepsilon }\) with \(q\equiv \varepsilon \pmod 3\). Moreover, if \(q\in \pi \), then the unique \(\pi \)Hall subgroup of \(D_{q1}\) is not \(\pi \)maximal in S. Indeed, this subgroup is a cyclic \(\pi _+\)Hall subgroup of S. Lemma 2.5 implies that this subgroup is conjugate to a subgroup in some Frobenius group \(C_q:C_{\frac{1}{2}(q1)}\le S\) and normalizes a Sylow qsubgroup of S. Hence, if \(2\notin \pi \), then H appears in Table 5.
Suppose that \(2\in \pi \). In order to show that H appears in Table 6, it is sufficient to show that a \(\pi \)Hall subgroup H of \(D_{q\varepsilon }\) is not \(\pi \)submaximal in S if \(\pi _\varepsilon =\{2\}\), \(3\in \pi \) and \(q\not \equiv \varepsilon \pmod 8\). Indeed, if \(q\equiv \varepsilon \pmod 8\), then \(H=(q\varepsilon )_2=2<S_2\) and \(H<P\) for some Sylow 2subgroup P of S, which contradicts Lemma 2.9; if \(q\equiv \pm 3\pmod 8\), then H is an elementary abelian Sylow 2subgroup of S, and \(H=4\), which contradicts Lemma 2.9 since, by the Sylow Theorem, H is contained as a normal subgroup in a maximal subgroup of S isomorphic to \(A_4\) and this subgroups is a \(\pi \)group.
Note that S has exactly one conjugacy class of cyclic subgroups of order \(q1\) (see the character table of S in [27, Theorem 13]). Hence, if \(2\notin \pi \), then any \(\pi \)Hall subgroup of \(E_q^{1+1}:C_{(q1)}\) is contained as a \(\pi \)Hall subgroup in some maximal subgroup of kind \({C_{(q1)_\pi }:C_4}\). Thus, in the case when \(2\notin \pi \), H appears in Table 7.
Now assume that \(2\in \pi \). If one of the sets \(\pi _0\), \(\pi _+\), \(\pi _\) is empty, then any \(\pi \)Hall subgroup of the respective subgroups: or \({C_{(q1)_\pi }:C_4}\), or \(D_{2(q r+1)}\), or \(D_{2(q r+1)}\), is a 2group, but is not a Sylow 2subgroup. Then Lemma 2.9 implies that case where H is a \(\pi \)Hall subgroup in one of these maximal subgroup of S is impossible. In the remaining cases, H appears in Table 8.
Case (I)(5) \(S=L_3(3)\). In this case, \(\pi (S)=\{2,3,13\}.\) By Lemma 2.1, U is one of the following groups: or \(E_{3^2}:GL_2(3)\), or \(C_{13}:C_{3}\), or \(S_4\). Since \(\pi (S)\nsubseteq \pi \) and \(\pi \cap \pi (S)>1\), the intersection \(\pi \cap \pi (S)\) coincides with one of sets \(\{3,13\}\), \(\{2,13\}\), and \(\{2,3\}\).
Suppose that \(\pi \cap \pi (S)=\{3,13\}\). Then a \(\pi \)Hall subgroup H of \(E_{3^2}:GL_2(3)\) coincides with a Sylow 3subgroup \(3_+^{1+2}\) of S and appears in Table 9. A \(\pi \)Hall subgroup H of \(C_{13}:C_{3}\) is \(C_{13}:C_{3}\) itself and H appears in Table 9. A \(\pi \)Hall subgroup H of \(S_4\) is of order 3 and Lemma 2.9 implies that this case is impossible.
Suppose that \(\pi \cap \pi (S)=\{2,13\}\). Then a \(\pi \)Hall subgroup H of \(E_{3^2}:GL_2(3)\) coincides with a semidihedral Sylow 2subgroup \(SD_{16}\) of both \(E_{3^2}:GL_2(3)\) and S, and H appears in Table 10. A \(\pi \)Hall subgroup H of \(C_{13}:C_{3}\) is cyclic of order 13 and appears in Table 10. A \(\pi \)Hall subgroup H of \(S_4\) is of order 8 and is not a Sylow 2subgroup of S. We may exclude this case by Lemma 2.9.
Finally, suppose that \(\pi \cap \pi (S)=\{2,3\}\). Then a \(\pi \)Hall subgroup H of \(E_{3^2}:GL_2(3)\) is \(E_{3^2}:GL_2(3)\) itself and H appears in Table 11. A \(\pi \)Hall subgroup H of \(C_{13}:C_{3}\) is cyclic of order 3 and Lemma 2.9 implies that this case is impossible. A \(\pi \)Hall subgroup H of \(S_4\) is \(S_4\) itself and it appears in Table 11. \(\square \)
4.2 The \(\pi \)submaximality, \(\pi \)maximality, conjugacy, and intravariancy for subgroups in Tables 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10
In this section, as in above, S is a group in the Thompson list \(\mathcal T\).
We prove that the converse of Proposition 4.1 holds, that is, the subgroups placed in Tables 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 are \(\pi \)submaximal under corresponding conditions. Moreover, we prove that the information about the \(\pi \)maximality, the conjugacy, the action of \({{\mathrm{Aut}}}(S)\) on the conjugacy classes of this subgroups, and the intravariancy in this tables is correct.
Proposition 4.2
 (1)
\(H\in {{\mathrm{sm}}}_\pi (S)\).
 (2)\(H\in {{\mathrm{Hall}}}_\pi (M)\) for every \(M\lessdot S\) such that \(H\le M\) (and so \(H\in {{\mathrm{m}}}_\pi (S)\)) with the exception of the following cases:
 (2a)\(S=L_2(q)\) for some prime q such that \(q>3\) and \(q^2+1\equiv 0\pmod 5\); \(2,3\in \pi \), \(H=D_{(q\varepsilon )_\pi }\), \(\varepsilon \in \{+,\}\), \(\pi \cap \pi (q\varepsilon )=\{2\}\) andIn this case, H is a Sylow 2subgroup of S, \(H\cong D_8\), and \(H<S_4\lessdot S\).$$\begin{aligned} q\equiv \varepsilon 7 \pmod {48}. \end{aligned}$$
 (2b)\(S=L_2(q)\) for some prime q such that \(q>3\) and \(q^2+1\equiv 0\pmod 5\); \({2,3\in \pi }\), \({H=D_{(q\varepsilon )_\pi }}\), \({\varepsilon \in \{+,\}}\), \({{\pi \cap \pi (q\varepsilon )}=\{2,3\}}\) andIn this case, \(H\cong S_3\cong D_6\), H is a \(\{2,3\}\)Hall subgroup of the normalizer in S of a Sylow 3subgroup and \(H<S_4\lessdot S\).$$\begin{aligned} {q\equiv \varepsilon 7,\varepsilon 31\pmod {72}}. \end{aligned}$$
 (2c)
\(S=L_3(3)\), \(\pi \cap \pi (S)=\{2,3\}\), and \({H={3_+^{1+2}:C_2^2}}\). In this case, \({H<{E_{3^2}:GL_2(3)}\lessdot S}\).
 (2d)
\(S=L_3(3)\), \(\pi \cap \pi (S)=\{2,3\}\), and \(H=GL_2(3)\). In this case, \({H<{E_{3^2}:GL_2(3)}\lessdot S}\).
 (2a)
 (3)S has a unique conjugacy class of \(\pi \)submaximal subgroups isomorphic to H with the exception of the following cases when S has exactly two conjugacy classes of \(\pi \)submaximal subgroups isomorphic to H that are fused in \({{\mathrm{Aut}}}(S)\):
 (3a)
\(S=L_2(q)\) where q is a prime such that \(q>3\), \(q^2+1\equiv 0\pmod 5\), \(q\equiv \pm 1\pmod 8\), \(2,3\in \pi \), and \(H=S_4\);
 (3b)
\(S=L_3(3)\), \(\pi \cap \pi (S)=\{2,3\}\), and \(H=E_{3^2}:GL_2(3)\).
 (3a)
 (4)
H is intravariant in S excepting the cases determined in (3) where S has two conjugacy classes of \(\pi \)submaximal subgroups isomorphic to H.
Proof
Firstly, we prove (2).
Non\(\pi \)maximality of H and corresponding inclusions in cases (2b) or (2c) follow from [2].
Suppose case (2a) holds. Then \(S=L_2(q)\), \(2,3\in \pi \), \(H=D_{(q\varepsilon )_\pi }\) where q is a prime such that \(q>3\) and \(q^2+1\equiv 0\pmod 5\), \(\varepsilon \in \{+,\}\).
Consider the case where \(\pi \cap \pi (q\varepsilon )=\{2\}\) and \(q\equiv \varepsilon 7 \pmod {48}\). Then \(q\equiv \varepsilon \pmod 8\). This means that H coincides with a Sylow 2subgroup of S and, moreover, S contains a maximal subgroup isomorphic to \(S_4\) by Lemma 2.1. Since \(q\equiv 7\varepsilon \pmod {16}\), we have \(H=(q\varepsilon )_\pi =8=S_4_2\). The Sylow Theorem implies that H is conjugate to a Sylow 2subgroup of \(S_4\). Thus, H is not \(\pi \)maximal in S in view of \(2,3\in \pi \) and \(S_4\) is a \(\pi \)group.
Now consider the case where \(\pi \cap \pi (q\varepsilon )=\{2,3\}\) and \(q\equiv \varepsilon 7,\varepsilon 31\pmod {72}\). Then \(q\equiv \varepsilon \pmod 8\). It means that S contains a maximal subgroup M isomorphic to \(S_4\) by Lemma 2.1. But it is easy to calculate that \(H=(q\varepsilon )_\pi =6\) and so \(H\cong S_3\). Moreover, H contains a Sylow 3subgroup P of S since \(S_3=3\). It follows that H is contained in \(N_S(P)\). By considering the maximal subgroups of S given in Lemma 2.1, it is easy to see that \(N_S(P)=D_{q\varepsilon }\), so H is a \(\{2,3\}\)Hall subgroup of \(N_S(P)\). By the Sylow Theorem, we can assume that \(P<M\). Since \(N_M(P)\cong S_3\), we obtain \(N_M(P)\in {{\mathrm{Hall}}}_{\{2,3\}}(N_S(P))\). The Hall Theorem and the solvability of \(N_S(P)\) imply that H is conjugate with \(N_M(P)\) and H is not \(\pi \)maximal in S since \(2,3\in \pi \) and \(S_4\) is a \(\pi \)group.
 (I)
up to isomorphism, there is a unique maximal subgroup M of S such that H divides M, \(H\le M\) and H is a \(\pi \)Hall subgroup of M.
 (II)
\(S=L_2(q)\) where \(q=2^p\) for a prime p, \(2\notin \pi \), \(H=C_{(q1)_\pi }\), and, up to isomorphism, there are exactly two maximal subgroup M of S such that H divides M, namely either \(M=E_q:C_{(q1)}\) or \(M=D_{2(q1)}\). In this case, H is a \(\pi \)Hall subgroup of every maximal subgroup containing H.
 (III)
\(S=L_2(q)\) where \(q=2^p\) for a prime p, \(2\in \pi \), \(H=D_{2(q1)_\pi }\), and, up to isomorphism, there are exactly two maximal subgroup M of S such that H divides M, namely either \(M=E_q:C_{(q1)}\) or \(M=D_{2(q1)}\). In this case H is not isomorphic to a subgroup of \(E_q:C_{(q1)}\) and is contained only in \(M=D_{2(q1)}\). Hence H is a \(\pi \)Hall subgroup of every maximal subgroup containing H.
 (IV)
\(S=L_2(q)\) where either \(q=3^p\) for odd prime or \(q>3\) is a prime, \(2\notin \pi \), \(H=C_{(q1)_\pi }\) and, up to isomorphism, there are exactly two maximal subgroup M of S such that H divides M, namely either \(M=E_q:C_{\frac{1}{2}(q1)}\) or \(M=D_{q1}\). In this case, H is a \(\pi \)Hall subgroup of every maximal subgroup containing H.
 (V)
\(S=Sz(q)\) where \(q=2^p\) for a prime p, \(2\notin \pi \), \(H=C_{(q1)_\pi }\), and, up to isomorphism, there are exactly two maximal subgroup M of S such that H divides M, namely either \(M=E^{1+1}_q:C_{(q1)}\) or \(M=C_{q1}:C_4\). In this case, H is a \(\pi \)Hall subgroup of every maximal subgroup containing H.
 (VI)
\(S=Sz(q)\) where \(q=2^p\) for a prime p, \(2\in \pi \), \(H=C_{(q1)_\pi }:C_4\), and, up to isomorphism, there are exactly two maximal subgroup M of S such that H divides M, namely either \(M=E^{1+1}_q:C_{(q1)}\) or \(M=C_{q1}:C_4\). In this case H is not isomorphic to a subgroup of \(E^{1+1}_q:C_{(q1)}\) and is contained only in \(M=C_{q1}:C_4\). Hence H is a \(\pi \)Hall subgroup of every maximal subgroup containing H.
 (VII)
\(S=L_2(q)\) where q is a prime such that \(q>3\) and \(q^2+1\equiv 0\pmod 5\), \(2\in \pi \), \(H=D_{(q\varepsilon )_\pi }\) for some \(\varepsilon \in \{+,\}\), H divides 24, and, up to isomorphism, there are exactly two maximal subgroup M of S such that H divides M, namely either \(M=D_{q\varepsilon }\) or \(M\in \{A_4,S_4\}\). This case will be separately considered below.
 (VIII)
\(S=L_2(q)\) where q is a prime such that \(q>3\) and \(q^2+1\equiv 0\pmod 5\), \(2,3\in \pi \), \(H\in \{A_4,S_4\}\), H divides \(q\varepsilon \) for some \(\varepsilon \in \{+,\}\) and, up to isomorphism, there are exactly two maximal subgroup M of S such that H divides M, namely either \(M=D_{q\varepsilon }\) or \(M\in \{A_4,S_4\}\). In this case H is not isomorphic to any subgroup of \(D_{q\varepsilon }\), so H itself is maximal and coincide with its \(\pi \)Hall subgroup.
 (IX)
\({S=L_3(3)}\), \({\pi =\{2,3\}}\), \({H=S_4}\), and, up to isomorphism, there are exactly two maximal subgroup M of S such that H divides M, namely either \({M=E_{3^2}:GL_2(3)}\) or \(M=S_4\). In this case, H itself is maximal and coincide with its \(\pi \)Hall subgroup.
Note that we have the condition in Table 6 for H that ether \(\pi _\varepsilon \ne \{2\}\), or \(3\notin \pi \), or \(q\equiv \varepsilon \pmod 8\).
Thus, we can consider that \(\pi _\varepsilon =\{2\}\). Suppose that \(3\notin \pi \). Since H divides 24, we have \(H=D_8\) or \(H=D_4=C_2\times C_2\). In the first case, \(M=S_4\) and H is a \(\pi \)Hall subgroup of both \(D_{q\varepsilon }\) and \(S_4\), and in the second case, \(M=A_4\) and H is \(\pi \)Hall in both \(D_{q\varepsilon }\) and \(A_4\).
This completes the proof of (2).
Now we prove (1). In view of (2), it is sufficient to prove that if one of the exceptional cases (2a)–(2d) in (2) holds, then \(H=K\cap S\) for some \(K\in {{\mathrm{m}}}_\pi (G)\) where \(G={{\mathrm{Aut}}}(S)\).
Suppose one of cases (2a)–(2b) holds. Then \(S=L_2(q)\), \(2,3\in \pi \), \(H=D_{(q\varepsilon )_\pi }\) where q is a prime such that \(q>3\), \(q^2+1\equiv 0\pmod 5\), \(\varepsilon \in \{+,\}\), and \(G=PGL_2(q)\). It follows from [1, Table 8.1] that \(H\le U\lessdot G\) where \(U\cong D_{2(q\varepsilon )}\). Let \(K\cong D_{2(q\varepsilon )_\pi }\) be a \(\pi \)Hall subgroup of U. Then \(G=SK\). Assume that \(K\notin {{\mathrm{m}}}_\pi (G)\). Then \(K<L\) for some \(\pi \)maximal subgroup L of G. It follows from \(\pi (S)\nsubseteq \pi \) that \(S\nleq L\). Let V be a maximal subgroup of G such that \(L\le V\). In view of Lemma 2.2, either \(V\cap S\lessdot S\) or \(q=7\) and \(V\cap S\in \{D_6, D_8\}\). It is easy to see that the the both arithmetic conditions \(q\equiv \varepsilon 7\pmod {48}\) in (2a) and \(q\equiv \varepsilon 7,\varepsilon 31\pmod {72}\) in (2b) imply that \(q\equiv \pm 1\pmod 8\), so S has no maximal subgroup isomorphic to \(A_4\). Moreover, \(VS=G=PGL_2(q)\). If we assume that \(V\cap S\lessdot S\) and \(V\cap S\cong S_4\), then the conjugacy class in S of maximal subgroups isomorphic to \(S_4\) would be invariant under \(G={{\mathrm{Aut}}}(S)\). But it is not so, see Table 14. Thus, \(V\cap S\) is one of the following groups: \(C_q:C_{\frac{1}{2}(q1)}\), \(D_{q+\varepsilon }\), and \(D_{q\varepsilon }\). Since \(V\le N_G(V\cap S)\), we have \(V=N_G(V\cap S)\) and V coincides with one of \(C_q:C_{q1}\), \(D_{2(q+\varepsilon )}\), and \(D_{2(q\varepsilon )}\). But V contains \(K\cong D_{2(q\varepsilon )_\pi }\) and \(K\notin {{\mathrm{Hall}}}_\pi (V)\). We can exclude the case where \(V=C_q:C_{q1}\) since in this case every dihedral subgroup of V is isomorphic to \(D_{2q}\) but \(((q\varepsilon )_\pi ,q)=1\). If \(V= D_{2(q+\varepsilon )}\), then \((q\varepsilon )_\pi \) divide \((q\varepsilon ,q+\varepsilon )=2\), but \((q\varepsilon )_\pi =H\in \{6,8\}\); a contradiction. The last case when \(V=D_{2(q\ \varepsilon )}\) is impossible in view of \(K\notin {{\mathrm{Hall}}}_\pi (V)\). Thus, \(K\in {{\mathrm{m}}}_\pi (G)\) and \(H=K\cap S\in {{\mathrm{sm}}}_\pi (S)\).
In the cases (2c) and (2d), \(\pi \cap \pi (S)=\{2,3\}\). Let \(G={{\mathrm{Aut}}}(S)\cong L_3(3):C_2\). Lemma 2.2 implies that \(H=V\cap S\) where \(V\lessdot G\) and \(V=GL_2(3):C_2\) in the case (2c) and \(V\cong 3_+^{1+2}:D_8\) in the case (2d). Thus \(V\in {{\mathrm{m}}}_\pi (G)\) and \(H\in {{\mathrm{sm}}}_\pi (S)\). Therefore, (1) is proved.
Prove (3). By Lemma 2.1 (Tables 14 and 16), if one of the exceptional cases (3a) or (3b) holds, then H is a maximal subgroup of corresponding S, any maximal subgroup of S which is not isomorphic to H does not contain a subgroup isomorphic to H, and S has exactly two conjugacy classes of (maximal) subgroups isomorphic to H interchanged by every noninner automorphism of S.
We need to prove that in the remaining cases, every \(K\in {{\mathrm{sm}}}_\pi (S)\) isomorphic to H is conjugate to H. By (2), if there is a maximal subgroup M of S containing both H and K, then \(H,K\in {{\mathrm{Hall}}}_\pi (M)\). The solvability of M implies that H and K are conjugate in M in view the Hall Theorem. Hence we can consider that H and K are contained in nonconjugate maximal subgroups M and N, respectively.
If M and N are isomorphic, then Lemma 2.1 implies that either (a) \(S=L_2(q)\), \(q>3\) is a prime, and \(M\cong N\cong S_4\), or (b) \(S=L_3(3)\) and \(M\cong N\cong E_{3^2}:GL_2(3)\). Since \(H\in {{\mathrm{Hall}}}_\pi (M)\) and \(K\in {{\mathrm{Hall}}}_\pi (N)\) in view of (2) and \(H\ne M\) and \(K\ne N\) (if not, one of the exceptional cases (3a) or (3b) holds), H and K are Sylow 2 or 3subgroups of M and N, respectively. Now Lemma 2.9 and \(H,K\in {{\mathrm{sm}}}_\pi (S)\) imply that H, K are Sylow subgroups in S and they are conjugate by the Sylow Theorem.
Consider the case where M and N are nonisomorphic maximal subgroups of S. In this case, \(H=K\) divides (M, N) and, by the arguments similar above, we can consider that the numbers \(H=K\) and (M, N) are not powers of primes. Consider all possibilities for S.
Let \(S=L_2(q)\) where \(q=2^p\), p is a prime. Since (M, N) is not a power of prime, by the information in Table 12 we can consider that \(M=E_q:C_{(q1)}\) and \(N=D_{2(q1)}\). If \(2\in \pi \), then the \(\pi \)Hall subgroups of M and N are nonisomorphic. Hence \(2\notin \pi \). In this case, both H and K are abelian \(\pi \)Hall subgroups of S and they are conjugate by Lemma 2.5.
Let \(S=L_2(q)\), where \(q=3^p\) and p is an odd prime. Note that the order of \(E_q:C_{\frac{1}{2}(q1)}\) is odd and the orders of \(D_{q1}\) and \(D_{q+1}\) are not divisible by 3. Now it easy to see from Table 13 that the condition that (M, N) is not a power of prime implies that we can consider that \(M=E_q:C_{\frac{1}{2}(q1)}\), \(N=D_{q1}\), and \(2\notin \pi \). In this case, both H and K are abelian \(\pi \)Hall subgroups of S and they are conjugate by Lemma 2.5.
Let \(S=L_2(q)\) where q is a prime such that \(q>3\) and \(q^2+1\equiv 0\pmod 5\). The case when one of M and N is \(E_q:C_{\frac{1}{2}(q1)}\) and other one is \(D_{q1}\) can be argued similarly as the previous case, and as (M, N) is not a power of prime, we can consider that \(M\in \{A_4,S_4\}\). Hence \(2,3\in \pi \). If \(M=A_4\), then \(H=M\) (if not, H is a power of 2 or 3), but the maximal subgroups of S of the other types contain no subgroups isomorphic to \(A_4\). Hence \(M=S_4\). Since the other maximal subgroups of S do not contain subgroups isomorphic to \(S_4\) and H is not a power of a prime, we have that \(H\cong S_3\). Since a maximal subgroup N of S contains a subgroup K isomorphic to \(H\cong S_3\), it is easy to see from Table 14 that \(N=D_{q\varepsilon }\) for some \(\varepsilon \in \{+,\}\) and in view of \(K\in {{\mathrm{Hall}}}_\pi (N)\) we have that \(S_3=N_3=3\). Hence N is the normalizer of a Sylow 3subgroup P of S and \(K\in {{\mathrm{Hall}}}_\pi (N_S(P))\). But \(H\cong K\) means that H is also a \(\pi \)Hall subgroup of some (solvable) normalizer of a Sylow 3subgroup Q of S. Hence by the Sylow and Hall Theorems, H and K are conjugate.
Let \(S=Sz(q)\), where \(q=2^p\), p is an odd prime. This case can be investigated without essential changes as the case \(S=L_2(q)\), \(q=2^p\).
Finally, let \(S=L_3(3)\). Since (M, N) is not a power of prime and in view of the information in Table 16, we can consider that \(M=E_{3^2}:GL_2(3)\), \(N=S_4\) and \(\pi =\{2,3\}\). But \(H\in {{\mathrm{Hall}}}_\pi (M)\) and \(K\in {{\mathrm{Hall}}}_\pi (N)\) implies that \(H=M\not \cong N=K\); a contradiction.
Statement (4) is a straightforward consequence of (3). \(\square \)
4.3 The pronormality of the \(\pi \)submaximal subgroups of minimal nonsolvable groups
In order to complete the proof of Theorem 1.1, we need to establish the pronormality of the \(\pi \)maximal subgroups in minimal simple groups, that is, we need to show that every subgroup H appearing in one of Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11 is pronormal in the corresponding group \(S\in \mathcal T\).
Proposition 4.3
Let S be a minimal simple group and \(H\in {{\mathrm{sm}}}_\pi (S)\). Then H is pronormal in S.
Proof
Let \(g\in S\). We need to show that H and \(H^g\) are conjugate in \(\langle H,H^g\rangle \). It is trivial if \(\langle H,H^g\rangle =S\). Hence we can consider that \(\langle H,H^g\rangle \le M\) for some \(M\lessdot S\). If \(H\in {{\mathrm{Hall}}}_\pi (M)\), then \(H,H^g\in {{\mathrm{Hall}}}_\pi (\langle H,H^g\rangle )\) and the solvability of M implies that H and \(H^g\) are conjugate in \(\langle H,H^g\rangle \).
 (2a)
\(S=L_2(q)\) where q is a prime such that \(q>3\) and \(q^2+1\equiv 0\pmod 5\), \(2,3\in \pi \), \(H=D_{(q\varepsilon )_\pi }\) \(\varepsilon \in \{+,\}\), \(\text {}\pi \cap \pi (q\varepsilon )=\{2\}\), \(q\equiv \varepsilon 7\pmod {48}\), and \(H\cong D_8\) is a Sylow 2subgroup of S.
 (2b)
\(S=L_2(q)\) where q is a prime such that \(q>3\) and \(q^2+1\equiv 0\pmod 5\), \(2,3\in \pi \), \(H=D_{(q\varepsilon )_\pi } \) \(\varepsilon \in \{+,\}\), \(\pi \cap \pi (q\varepsilon )=\{2,3\}\), \(q\equiv \varepsilon 7,\varepsilon 31\pmod {72}\), and \(H\cong S_3\cong D_6\) is a \(\{2,3\}\)Hall subgroup of the normalizer of a Sylow 3subgroup of S.
 (2c)
\(S=L_3(3)\), \(\pi \cap \pi (S)=\{2,3\}\), and \(H=3_+^{1+2}:C_2^2\) is the normalizer of a Sylow 3subgroup of S.
 (2d)
\(S=L_3(3)\), \(\pi \cap \pi (S)=\{2,3\}\), and \(H=GL_2(3)\).
Proof of Corollary 1.1. The corollary is a straightforward consequence of Propositions 1.2(4) and 4.3.
Footnotes
 1.
 2.
This problem is one of ten open problems in this talk [38]. Another problem [38, Frage (i)] is the following conjecture: a subgroup A of a group G is subnormal if \(N_A(H\cap A)/(H\cap A)\) is \(\pi '\)group for every set \(\pi \) and every \(H\in {{\mathrm{m}}}_\pi (G)\). This is a converse statement to the strong version of the Wielandt–Hartley Theorem (see previous remark) and it was proved by P. Kleidman in his famous work [17].
 3.
In Wielandt’s terminology, a subgroup H of a group G is said to be intravariant if the conjugacy class of H in G is \({{\mathrm{Aut}}}(G)\)invariant, that is, for any \(\alpha \in {{\mathrm{Aut}}}(G)\), there is \(g\in G\) such that \(H^\alpha =H^g\). Recall, a subgroup H of a group G is said to be pronormal if H and \(H^g\) are conjugate in \(\langle H,H^g\rangle \) for any \(g\in G\).
Notes
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