# The length and depth of algebraic groups

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## Abstract

Let *G* be a connected algebraic group. An unrefinable chain of *G* is a chain of subgroups \(G = G_0> G_1> \cdots > G_t = 1\), where each \(G_i\) is a maximal connected subgroup of \(G_{i-1}\). We introduce the notion of the length (respectively, depth) of *G*, defined as the maximal (respectively, minimal) length of such a chain. Working over an algebraically closed field, we calculate the length of a connected group *G* in terms of the dimension of its unipotent radical \(R_u(G)\) and the dimension of a Borel subgroup *B* of the reductive quotient \(G/R_u(G)\). In particular, a simple algebraic group of rank *r* has length \(\dim B + r\), which gives a natural extension of a theorem of Solomon and Turull on finite quasisimple groups of Lie type. We then deduce that the length of any connected algebraic group *G* exceeds \(\frac{1}{2} \dim G\). We also study the depth of simple algebraic groups. In characteristic zero, we show that the depth of such a group is at most 6 (this bound is sharp). In the positive characteristic setting, we calculate the exact depth of each exceptional algebraic group and we prove that the depth of a classical group (over a fixed algebraically closed field of positive characteristic) tends to infinity with the rank of the group. Finally we study the chain difference of an algebraic group, which is the difference between its length and its depth. In particular we prove that, for any connected algebraic group *G* with soluble radical *R*(*G*), the dimension of *G* / *R*(*G*) is bounded above in terms of the chain difference of *G*.

## Mathematics Subject Classification

Primary 20E32 20E15 Secondary 20G15 20E28## 1 Introduction

*length*of a finite group

*G*, denoted by

*l*(

*G*), is the maximum length of a chain of subgroups of

*G*. This interesting invariant was the subject of several papers by Janko and Harada [10, 13, 14] in the 1960s, culminating in Harada’s description of the finite simple groups of length at most 7 in [10]. In more recent years, the notion of length has arisen naturally in several different contexts. For example, Babai [1] considered the length of the symmetric group \(S_n\) in relation to the computational complexity of algorithms for finite permutation groups (a precise formula for \(l(S_n)\) was later determined by Cameron, Solomon and Turull in [6]). Motivated by applications to fixed-point-free automorphisms of finite soluble groups, Seitz, Solomon and Turull studied the length of finite groups of Lie type in a series of papers in the early 1990s [21, 23, 24]. Let us highlight one of their main results, [24, Theorem A*], which states that if \(G=G_r(p^k)\) is a finite quasisimple group of Lie type and

*k*is sufficiently large (with respect to the characteristic

*p*), then

*B*is a Borel subgroup of

*G*and

*r*is the twisted Lie rank of

*G*.

*depth*of a finite group

*G*, denoted by \(\lambda (G)\), is the minimal length of a chain of subgroups

*G*) and [4, Theorem 2] shows that alternating groups have bounded depth (more precisely, \(\lambda (A_n) \leqslant 23\) for all

*n*, whereas \(l(A_n)\) tends to infinity with

*n*). Upper bounds on the depth of each simple group of Lie type over \(\mathbb {F}_q\) are presented in [4, Theorem 4]; the bounds are given in terms of

*k*, where \(q=p^k\) with

*p*a prime.

*G*be a connected algebraic group over an algebraically closed field of characteristic \(p \geqslant 0\). An

*unrefinable*chain of length

*t*of

*G*is a chain of subgroups

*length*of

*G*, denoted by

*l*(

*G*), to be the maximal length of an unrefinable chain. Similarly, the

*depth*\(\lambda (G)\) of

*G*is the minimal length of such a chain. Notice that we impose the condition that the subgroups in an unrefinable chain are connected, which seems to be the most natural (and interesting) definition in this setting.

In the statements of our main results, and for the remainder of the paper, we assume that the given algebraic group is connected and the underlying field is algebraically closed (unless stated otherwise). Also note that our results are independent of any choice of isogeny type. Our first result concerns the length of an algebraic group.

### Theorem 1

*G*be an algebraic group and let

*B*be a Borel subgroup of the reductive group \(\bar{G} = G/R_u(G)\). Then

*r*is the semisimple rank of \(\bar{G}\).

### Corollary 2

*G*be a simple algebraic group of rank

*r*and let

*B*be a Borel subgroup of

*G*. Then

*G*of maximum length includes a maximal parabolic subgroup.

The last sentence of the corollary is justified in Remark 3.1.

By Lemma 2.2, the solubility of *B* implies that \(l(B) = \dim B\), so Corollary 2 is the algebraic group analogue of the aforementioned result of Solomon and Turull [24, Theorem A*] for finite quasisimple groups (see (1) above).

Next, we relate the length of arbitrary algebraic groups *G* to their dimension. We clearly have \(l(G) \leqslant \dim G\).

### Theorem 3

*G*be an algebraic group.

- (i)
\(l(G) > \frac{1}{2}\dim G\).

- (ii)
\(l(G) = \dim G\) if and only if \(G/R(G) \cong A_1^t\) for some \(t \geqslant 0\), where

*R*(*G*) is the soluble radical of*G*.

### Theorem 4

*G*be a simple algebraic group in characteristic zero. Then

*p*, define a sequence \(e_n(p)\) (\(n\in \mathbb {N}\)) as follows: \(e_1(p)=p\), and for \(l>1\),

### Theorem 5

*G*be a simple algebraic group in characteristic \(p>0\) with rank

*r*.

- (i)
If

*G*is an exceptional group then \(\lambda (G) \leqslant 9\), with equality if and only if \(G=E_8\) and \(p=2\). - (ii)
If

*G*is a classical group, then \(\lambda (G) \leqslant 2(\log _2r)^2+12\). - (iii)
For any

*G*, we have \(\lambda (G) \geqslant \psi _p(r)\). In particular, \(\lambda (G)\rightarrow \infty \) as \(r\rightarrow \infty \).

Part (i) of Theorem 5 is an immediate corollary of Theorem 3.3, which gives the exact depth of each exceptional algebraic group. For parts (ii) and (iii), see Theorems 3.4 and 3.10, respectively. We also give an example (Example 3.11) to show that the lower bound \(\psi _p(r)\) in (iii) is of roughly the correct order of magnitude.

By a well-known theorem of Iwasawa [12], the length and depth of a finite group *G* are equal if and only if *G* is supersoluble. This result does not extend directly to algebraic groups. However, it follows from our results on length and depth that the only simple algebraic group with \(\lambda (G)=l(G)\) is \(G = A_1\) (see Lemma 3.12). More generally, we prove the following result on arbitrary connected groups with this property, which can be viewed as a partial analogue of Iwasawa’s theorem.

### Theorem 6

Let *G* be an algebraic group satisfying \(\lambda (G)=l(G)\) and let *R*(*G*) be the radical of *G*. Then either *G* is soluble, or \(G/R(G) \cong A_1\).

In fact it follows from our arguments that \(\lambda (G) = l(G)\) if and only if \(\lambda (G) = \dim G\).

This is proved in Sect. 3.7. Note that the converse is false: for example, if \(G=UA_1\), a semidirect product where *U* is a nontrivial irreducible \(KA_1\)-module, then \(A_1\) is maximal in *G*, so \(\lambda (G)\leqslant 1+\lambda (A_1)=4\), while \(l(G) = \dim U+l(A_1)>4\). On the other hand, if \(G = U\times A_1\), or if *G* is a nonsplit extension of the irreducible module *U* by \(A_1\), then \(\lambda (G)=l(G)\).

*chain difference*of

*G*, which is defined by

### Theorem 7

*G*be an algebraic group and set \(\bar{G} = G/R(G)\). Then

This result will be proved in Sect. 3.8.

We also consider the *chain ratio*\(\mathrm{cr}(G) = l(G)/\lambda (G)\) of an algebraic group *G*, and show in Sect. 4 that in contrast to the chain difference, the dimension of *G* / *R*(*G*) is not in general bounded in terms of \(\mathrm{cr}(G)\).

## 2 Preliminaries

As stated in Sect. 1, for the remainder of the paper we assume *G* is a connected algebraic group over an algebraically closed field (unless stated otherwise). We start with the following elementary observation.

### Lemma 2.1

*G*be an algebraic group and let

*N*be a connected normal subgroup.

- (i)
\(\lambda (G) \leqslant l(G) \leqslant \dim G\).

- (ii)
\(l(G) = l(N)+l(G/N)\).

- (iii)
\(\lambda (G/N) \leqslant \lambda (G) \leqslant \lambda (N) + \lambda (G/N)\).

### Proof

Parts (i) and (iii) are obvious, and part (ii) is proved just as [6, Lemma 2.1]. \(\square \)

Recall that if *U* is a connected unipotent algebraic group, then the *Frattini subgroup*\(\Phi (U)\) of *U* is the intersection of the closed subgroups of *U* of codimension 1 (see [9]).

### Lemma 2.2

If *G* is a soluble algebraic group, then \(\lambda (G)= l(G) = \dim G\).

### Proof

It is sufficient to show that any connected maximal subgroup *M* of *G* has codimension 1. Write \(G = UT\), where \(U = R_u(G)\) and *T* is a maximal torus. If \(U\leqslant M\) then \(M=US\), where *S* is a connected maximal subgroup of *T*, and the result follows since \(\dim S = \dim T - 1\). Now assume \(U \not \leqslant M\), so \(M = (M \cap U)T\) and \(M \cap U\) is a maximal *T*-invariant subgroup of *U*. Now \(\Phi (U)\leqslant M\), so by factoring out \(\Phi (U)\) we can assume that \(\Phi (U)=1\). Then \(U \cong K^n\), an *n*-dimensional vector space over the underlying algebraically closed field *K* (see [9, Proposition 1]). Moreover, *T* acts linearly on *U*, and since *T* is diagonalisable on *V*, a maximal *T*-invariant subspace has codimension 1 (this is proved in much greater generality in [18, Theorem B]). Hence *M* has codimension 1 in *G* in this case also. \(\square \)

### Lemma 2.3

Let *G* be an algebraic group and let \(m \in \{1,2,3\}\). Then \(\lambda (G) = m\) if and only if \(\dim G = m\).

### Proof

The case \(m=1\) is obvious, so assume \(m \in \{2,3\}\). If \(\lambda (G)=2\) then *G* has a maximal \(T_1\) or \(U_1\) subgroup and clearly *G* is soluble, so \(\dim G = 2\) by Lemma 2.2. Conversely, if \(\dim G = 2\) then *G* is soluble and once again the result follows from Lemma 2.2.

*G*is soluble, then Lemma 2.2 implies that \(\lambda (G)=3\), otherwise \(G=A_1\) and \(\lambda (G)=3\) since

*k*, and similarly \(T_k\) for a

*k*-dimensional torus). Finally, suppose \(\lambda (G)=3\) and assume that

*G*is insoluble. If

*G*is reductive, it has a maximal connected subgroup of depth 2, hence of dimension 2, and the only possibility is \(G = A_1\). Otherwise, let \(U = R_u(G)\) be the unipotent radical of

*G*. Since

*G*is insoluble, the previous sentence implies that \(G/U \cong A_1\). But

*G*has a maximal subgroup of depth 2, which must be soluble of dimension 2. This is clearly not possible. \(\square \)

### Remark 2.4

Notice that the conclusion of Lemma 2.3 does not extend to integers \(m>3\). For example, if \(r \geqslant 2\) then the symplectic group \(C_r\) has a maximal \(A_1\) subgroup in characteristic 0, so there are depth 4 algebraic groups of arbitrarily large dimension.

### Lemma 2.5

Let \(G = U.L\) be an algebraic group, where *U* and *L* are nontrivial connected subgroups of *G*, with *U* normal. Then \(\lambda (G) \geqslant 1+\lambda (L)\).

### Proof

*i*, since otherwise the depth of

*L*would be less than

*t*. This means that

*i*minimal such that \(U \not \leqslant G_i\), we have \(G_i< G_iU < G_{i+1}\), contradicting the unrefinability of (2). \(\square \)

## 3 Proofs

### 3.1 Proof of Theorem 1

Let *G* be an algebraic group. The proof goes by induction on \(\dim G\). For \(\dim G = 1\), the result is obvious.

*G*is reductive. Write \(G = G_1\cdots G_tZ\), a commuting product with each \(G_i\) simple and \(Z=Z(G)^0\), and let \(B_i\) be a Borel subgroup of \(G_i\). Note that \(B = B_1\cdots B_tZ\) is a Borel subgroup of

*G*. Let \(r_i\) be the rank of \(G_i\), so \(r=\sum _{i}r_i\) is the semisimple rank of

*G*. By Lemma 2.1(ii) we have

*i*, and hence

*G*is simple of rank

*r*. By considering an unrefinable chain passing through

*B*, noting that \(l(B) = \dim B\) by Lemma 2.2, it follows that

*M*be a maximal connected subgroup of

*G*with \(l(M) = l(G)-1\). By [2, Corollary 3.9],

*M*is either parabolic or reductive. Suppose first that

*M*is reductive and let \(B_M\) be a Borel subgroup of

*M*. By induction,

*M*is a maximal parabolic subgroup.

*L*is a Levi subgroup. By induction,

*L*. Since \(B = QB_L\) is a Borel subgroup of

*G*, and \(\mathrm{rank}(L') = r-1\), it follows that \(l(G) = l(M)+1 = \dim B+ r\), as required. This completes the proof of Theorem 1.

### 3.2 Proof of Theorem 3

*G*is soluble, so assume otherwise. Write \(G/R(G) = G_1 \cdots G_t\), where each \(G_i\) is simple. By applying Corollary 2, it is easy to see that \(l(G_i) > \frac{1}{2}\dim G_i\) for each

*i*, so by combining Lemmas 2.1(ii) and 2.2 we get

### 3.3 Proof of Theorem 4

Let *G* be a simple algebraic group over an algebraically closed field of characteristic 0. The maximal connected subgroups of *G* were determined by Dynkin [7, 8] and we repeatedly apply these results throughout the proof. To begin with, let us assume *G* is a classical group of rank *r*.

*G*). One checks that \(\dim M > 3\) for every maximal connected subgroup

*M*of

*G*, so Lemma 2.3 implies that \(\lambda (G) \geqslant 5\). In fact, we see that equality holds since

*G*does not have a maximal \(A_1\) subgroup. But

*G*does have an unrefinable chain of length 5:

*G*has a maximal \(A_1\) subgroup if and only if \(r=2\), so we get \(\lambda (G)=4\) if \(r=2\), otherwise \(\lambda (G) \geqslant 5\). It is easy to see that \(\lambda (G)=5\) if \(r \geqslant 3\) and \(r \ne 6\). Indeed, we have the following unrefinable chains:

*M*be a maximal connected subgroup of

*G*. By [8], either \(M=B_3\) or

*M*is a parabolic subgroup of the form \(U_6A_5T_1\), \(U_{10}A_4A_1T_1\) or \(U_{12}A_3A_2T_1\) (here \(U_k\) denotes a normal unipotent subgroup of dimension

*k*). If

*M*is parabolic, then Lemma 2.1(iii) implies that \(\lambda (M) \geqslant \lambda (A_k)\) for some \(k \in \{3,4,5\}\) and thus \(\lambda (M) \geqslant 5\) by our above work. Since \(\lambda (B_3)=5\), the claim follows.

*G*is an exceptional group. By [7],

*G*has a maximal \(A_1\) subgroup if and only if \(G \ne E_6\), so \(\lambda (G)=4\) in these cases. For \(G=E_6\) we have \(\lambda (G) \geqslant 5\) and equality holds since

*G*has a maximal \(G_2\) subgroup (see [7]) and so there is an unrefinable chain

### 3.4 Proof of Theorem 5(i)

Let *G* be a simple algebraic group of rank *r* over an algebraically closed field *K* of characteristic \(p > 0\). In this subsection we determine the precise depth of *G* in the case where *G* is of exceptional type.

*c*in a column occurs in the row corresponding to \(p=\ell \), then \(\lambda (G) = c\) for all \(p \geqslant \ell \). For example, Table 1 indicates that

### Lemma 3.2

Let *G* be a simple algebraic group of rank \(r \leqslant 4\) in characteristic \(p > 0\). Then \(\lambda (G)\) is given in Table 1.

The depth of low rank simple algebraic groups

| \(A_1\) | \(A_2\) | \(B_2\) | \(G_2\) | \(A_3\) | \(B_3\) | \(C_3\) | \(A_4\) | \(B_4\) | \(C_4\) | \(D_4\) | \(F_4\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|

2 | 3 | 6 | 5 | 5 | 6 | 6 | 6 | 9 | 7 | 7 | 7 | 8 |

3 | 4 | 5 | 5 | 5 | 6 | 5 | 6 | 5 | 6 | 7 | 6 | |

5 | 4 | 5 | 6 | 5 | 5 | 5 | 6 | 5 | 6 | |||

7 | 4 | 5 | 4 | 5 | 6 | 5 | ||||||

11 | 4 | 4 | 5 | |||||||||

> 11 | 4 |

### Proof

*M*is either parabolic or reductive. If

*G*is an exceptional group, then the possibilities for

*M*have been determined by Liebeck and Seitz (see [17, Corollary 2(ii)]). Similarly, if

*G*is a classical group with natural module

*V*, then [16, Theorem 1] implies that either

*M*stabilises a proper nontrivial subspace of

*V*, or a tensor product decomposition of the form \(V = U \otimes W\), or

*M*is simple and \(V|_{M}\) is an irreducible

*KM*-module with

*p*-restricted highest weight. We refer the reader to [25, Table 5] for a convenient list of the relevant reductive maximal connected subgroups of

*G*. It will be useful to observe that \(\lambda (M) \geqslant 5\) if

*M*is a maximal parabolic subgroup of

*G*(this follows immediately from Lemma 2.5).

Suppose \(G = A_2\). If \(p \geqslant 3\) then *G* has a maximal \(A_1\) subgroup, so \(\lambda (G)=4\). On the other hand, if \(p=2\) then \(M = U_2A_1T_1\) and \(M_1 \in \{A_1T_1, U_2A_1\}\) are the only possibilities, so \(\lambda (G) = 2+\lambda (M_1) = 6\).

*M*is parabolic. Since \(\lambda (B_2) = 5\) as above, and \(\lambda (M) \geqslant 5\) when

*M*is parabolic, it follows that \(\lambda (G) = 6\).

Next suppose \(G = B_3\), so \(\lambda (G) \geqslant 5\) since \(\dim M > 3\). Now *G* has a maximal \(G_2\) subgroup, so \(\lambda (G) \leqslant \lambda (G_2)+1\) and thus \(\lambda (G)=5\) if \(p \geqslant 7\), otherwise \(\lambda (G) \leqslant 6\). By considering the various possibilities for *M*, it is easy to check that \(\lambda (M) \geqslant 5\) when \(p \in \{2,3,5\}\) (we can assume *M* is reductive, so the possibilities are listed in [25, Table 5]) and we conclude that \(\lambda (G)=6\). The case \(G = C_3\) is similar. If \(p \geqslant 7\) then \(M = A_1\) and \(\lambda (G)=4\). Now assume \(p \in \{2,3,5\}\), so \(\lambda (G) \geqslant 5\). If \(p \in \{3,5\}\) then we can take \(M = A_1A_1\), which gives \(\lambda (G)=5\). Finally, if \(p=2\) then one checks that \(\lambda (M) \geqslant 5\), with equality if \(M = G_2\), hence \(\lambda (G)=6\).

*M*is a parabolic subgroup, whence \(\lambda (G)=6\) since \(\lambda (B_2)=5\) as above. Finally, suppose \(p=2\). Here every maximal connected subgroup of

*G*is parabolic, so we need to consider the depth of \(P_1 = U_4A_3T_1\) and \(P_2 = U_6A_2A_1T_1\). The Levi factor of \(P_1\) is maximal, so \(P_1> A_3T_1 > A_3\) is unrefinable and thus \(\lambda (P_1) \leqslant 8\) since \(\lambda (A_3) = 6\). Now Lemma 2.5 gives \(\lambda (P_1) \geqslant 2+\lambda (A_3)=8\) and \(\lambda (P_2) \geqslant 3+\lambda (A_2) = 9\), so \(\lambda (P_1) = 8\) and \(\lambda (G)=9\).

*M*is either a parabolic subgroup, or \(M=B_3\), \(A_1B_2\) (\(p=3\)), \(A_1^4\) or \(A_2\) (\(p=2\)). Note that \(\lambda (B_3)=6\) and \(\lambda (A_2)=6\) (with \(p=2\) in the latter case). By applying Lemma 2.5, it is also easy to see that \(\lambda (M) \geqslant 6\) in the remaining cases. For example, if \(M = U_6A_3T_1\) then \(\lambda (M) \geqslant 1+\lambda (A_3T_1) \geqslant 7\). This justifies the claim.

*M*. If

*M*is reductive, then \(M = D_4\), \(A_1B_3\) or \(B_2B_2\). By our earlier work, \(\lambda (D_4)=7\) and \(\lambda (A_1B_3) \geqslant 1+\lambda (B_3) =7\). Similarly, \(\lambda (B_2B_2) =6\) and one checks that \(\lambda (M) \geqslant 6\) if

*M*is parabolic. The claim follows.

*M*of

*G*and thus \(\lambda (G)=6\). Finally, suppose \(p=2\). Here \(\lambda (G) \leqslant 7\) via the chain

*M*of

*G*. If

*M*is reductive, then \(M = C_2C_2\), \(A_1C_3\) or \(D_4\). By combining Lemma 2.5 with our earlier work, we have \(\lambda (A_1C_3) \geqslant 1+\lambda (C_3)=7\) and \(\lambda (D_4)=7\). It is also easy to see that \(\lambda (C_2C_2) = 6\). It is routine to verify the claim when

*M*is parabolic. For instance, if \(M = U_{10}A_3T_1\) then \(\lambda (M) \geqslant 1+\lambda (A_3T_1) \geqslant 2+\lambda (A_3)= 8\).

*G*has a maximal \(A_1\) subgroup if and only if \(p \geqslant 13\), so we may assume \(p<13\). If \(p \in \{7,11\}\) then \(\lambda (G)=5\) via the chains

*M*and using Lemma 2.5 and our earlier work, it is easy to show that \(\lambda (M) \geqslant 5\) and thus \(\lambda (G)=6\). Finally, let us assume \(p = 2\). First observe that \(\lambda (G) \leqslant 8\) via the chain

*M*of

*G*. If

*M*is reductive, then \(M= C_4\), \(B_4\) or \(A_2\tilde{A}_2\). As above, we have \(\lambda (B_4) = \lambda (C_4) = 7\) and \(\lambda (A_2\tilde{A}_2) \geqslant 1 + \lambda (A_2) = 7\) and the result follows. If \(M = UL\) is a parabolic subgroup, with unipotent radical

*U*, then Lemma 2.5 gives \(\lambda (M) \geqslant 1+\lambda (L)\) and it is straightforward to see that \(\lambda (L) \geqslant 6\). For example, if \(M = U_{20}A_1A_2T_1\) then Lemma 2.5 yields \(\lambda (L) \geqslant 2+\lambda (A_2) = 8\). The result follows. \(\square \)

We are now in a position to prove our main result for exceptional groups. In particular, part (i) of Theorem 5 is an immediate corollary of the following result. In Table 2, we adopt the same conventions as in Table 1.

### Theorem 3.3

Let *G* be an exceptional algebraic group in characteristic \(p > 0\). Then \(\lambda (G)\) is given in Table 2.

The depth of exceptional algebraic groups

| \(G_2\) | \(F_4\) | \(E_6\) | \(E_7\) | \(E_8\) |
---|---|---|---|---|---|

2 | 5 | 8 | 6 | 8 | 9 |

3 | 5 | 6 | 6 | 7 | 7 |

5 | 5 | 6 | 5 | 5 | 5 |

7 | 4 | 5 | 5 | 5 | |

11 | 5 | 5 | 5 | ||

13 | 4 | 5 | 5 | ||

17 | 4 | 5 | |||

19 | 5 | ||||

> 19 | 4 |

### Proof

*M*are determined in [17].

*G*has a maximal \(A_2\) subgroup and \(\lambda (A_2) = 4\) by Lemma 3.2, so \(\lambda (G)=5\). Now assume \(p \in \{2,3\}\). Here \(G_2<G\) is maximal and thus \(\lambda (G) \leqslant \lambda (G_2)+1 = 6\). We claim that \(\lambda (G)=6\). If

*M*is reductive, then [17] implies that

*M*is parabolic. This justifies the claim.

*M*be a maximal connected subgroup of

*G*. Suppose

*M*is reductive, in which case

*M*is parabolic and the claim follows.

*M*of

*G*, whence \(\lambda (G)=8\).

*G*has a maximal \(A_1\) subgroup. If \(5 \leqslant p \leqslant 19\) then \(\lambda (G)=5\) via the chain

*M*is reductive then [17, Corollary 2(ii)] implies that

*M*is parabolic.

*M*is reductive then (4) holds and we consider each possibility in turn. The bound is clear if \(M = A_1E_7\), \(A_4A_4\) or \(G_2F_4\) since \(\lambda (E_7) = \lambda (F_4)=8\) and \(\lambda (A_4)=9\). Similarly, \(\lambda (A_2E_6) \geqslant 8\) since \(\lambda (A_2) = \lambda (E_6)=6\). If \(M = A_8\) and \(M_1\) is reductive, then \(M_1=A_2A_2\) is the only option and thus \(\lambda (M_1) \geqslant 1+\lambda (A_2) = 7\). It is easy to see that \(\lambda (M_1) \geqslant 7\) if \(M_1\) is parabolic, so \(\lambda (A_8) \geqslant 8\) as required. Similarly, one checks that \(\lambda (D_8) \geqslant 8\). Finally, if

*M*is a maximal parabolic subgroup with Levi factor

*L*, then Lemma 2.5 gives \(\lambda (M) \geqslant 2+\lambda (L')\) and it is straightforward to show that \(\lambda (L') \geqslant 6\). The result follows. \(\square \)

### 3.5 Proof of Thereom 5(ii)

Now assume *G* is a simple classical algebraic group. The next result establishes the upper bound on \(\lambda (G)\) in part (ii) of Theorem 5.

### Theorem 3.4

*G*is a simple classical algebraic group of rank

*r*, then

We partition the proof into a sequence of lemmas, starting with the case where *G* is a symplectic group. Note that \(\lambda (B_r) = \lambda (C_r)\) when \(p=2\).

### Lemma 3.5

### Proof

*k*. We will repeatedly use the fact that if

*H*is a symplectic group with natural module

*V*, then the stabiliser in

*H*of any proper nondegenerate subspace of

*V*is a maximal connected subgroup of

*H*.

*M*be the stabiliser in

*G*of a nondegenerate

*r*-space, so \(M = C_{2^{a_1-1}}C_{2^{a_1-1}}\). Now

*M*has a diagonally embedded maximal subgroup of type \(C_{2^{a_1-1}}\), so there is an unrefinable chain

*G*to \(C_1\) in \(2a_1\) steps and thus

*M*be the stabiliser in

*G*of a nondegenerate \(2^{a_1}\)-space, so \(M = C_{2^{a_1}}C_{2^{a_2}+\cdots + 2^{a_k}}\). By Lemma 2.1, we have

### Lemma 3.6

If \(G=A_r\), then \(\lambda (G) \leqslant 2(\log _2r)^2+4\).

### Proof

*r*is odd then \(C_{(r+1)/2}\) is a maximal connected subgroup of

*G*, so Lemma 3.5 implies that

*r*is even then

### Lemma 3.7

Suppose \(G=D_r\), where \(r \geqslant 3\). Then \(\lambda (G) \leqslant 2(\log _2r)^2+11\).

### Proof

*r*is even and write \(r=2^{a_1}+\cdots + 2^{a_k}\), where \(a_1>a_2> \cdots > a_k \geqslant 1\). We claim that the upper bound on \(\lambda (G)\) in (5) holds, in which case

*k*. Note that if

*M*is the connected component of the stabiliser in

*G*of a nondegenerate \(\ell \)-space of the natural module, with \(1 \leqslant \ell \leqslant r\) and \(\ell \ne 2\), then

*M*is a maximal connected subgroup of

*G*(if \(\ell =2\) then \(M = T_1D_{r-1}\) is the Levi factor of a parabolic subgroup of

*G*).

*r*is even). Then \(M = D_{2^{a_1}}D_{2^{a_2}+\cdots +2^{a_k}}\) is a maximal closed connected subgroup of

*G*, so

*G*, so we can assume \(r \geqslant 7\). Let

*M*be the connected component of the stabiliser in

*G*of a nondegenerate 6-space. Then \(M = D_3D_{r-3}\) is a maximal connected subgroup of

*G*, so by the previous result for even rank, we get

The next lemma completes the proof of Theorem 3.4.

### Lemma 3.8

Suppose \(G=B_r\), where \(r \geqslant 3\). Then \(\lambda (G) \leqslant 2(\log _2r)^2+12\).

### Proof

This is an immediate corollary of Lemma 3.7 since \(D_r\) is a maximal connected subgroup of *G*. \(\square \)

### 3.6 Proof of Theorem 5(iii)

*r*over an algebraically closed field

*K*of characteristic \(p>0\), with natural module

*V*. If

*M*is a maximal connected subgroup of

*G*, then by [16, Theorem 1], one of the following holds:

- (i)
*M*is the connected stabiliser of a proper nontrivial subspace*U*of*V*that is either totally singular, nondegenerate, or a nonsingular 1-space (the latter only when*G*is orthogonal and \(p=2\)); - (ii)
*M*is the connected stabiliser \(Cl(U)\otimes Cl(W)\) of a tensor product decomposition \(V = U\otimes W\); - (iii)
\(M \in \mathcal {S}(G)\), the collection of maximal connected simple subgroups of

*G*such that*V*is a*p*-restricted irreducible*KM*-module.

### Lemma 3.9

Let *G* be as above, let \(M \in \mathcal {S}(G)\) and suppose *M* is of classical type. Then \(\mathrm{rank}(M) > \sqrt{\log _p r}\).

### Proof

Let \(k=\mathrm{rank}(M)\). Using Weyl’s character formula, it is easy to see that the *p*-restricted irreducible *KM*-module of largest dimension is the Steinberg module, which has dimension \(p^N\), where *N* is the number of positive roots in the root system of *M*. Since \(N\leqslant k^2\), it follows that \(\dim V \leqslant p^{k^2}\). The conclusion follows, as \(r<\dim V\). \(\square \)

### Theorem 3.10

If *G* is a simple algebraic group of rank *r* in characteristic \(p>0\), then \(\lambda (G) \geqslant \psi _p(r)\).

### Proof

The proof proceeds by induction on *r*. If \(r<e_3(p) = p^{p^{2p^2}}\), then \(\psi _p(r) \leqslant 2\), so the conclusion holds.

*G*is classical. Choose a maximal connected subgroup

*M*of

*G*such that \(\lambda (M)=\lambda (G)-1\). Then

*M*is as in one of the possibilities (i)-(iii) above, and in case (iii) we have \(\mathrm{rank}(M) > \sqrt{\log _p r}\), by Lemma 3.9. In cases (i) and (ii),

*M*has a simple quotient

*Cl*(

*U*) with \(\dim U \geqslant \sqrt{\dim V}\). Hence in any case, there is a simple connected group

*H*of rank at least \(\sqrt{\log _p r}\), such that \(\lambda (M)\geqslant \lambda (H)\). By induction,

The proof of Theorem 5 is now complete.

We conclude with an example showing that the lower bound \(\psi _p(r)\) in Theorem 5(iii) is of roughly the correct order of magnitude.

### Example 3.11

### 3.7 Proof of Theorem 6

We begin by classifying the simple algebraic groups *G* with \(\lambda (G)=l(G)\).

### Lemma 3.12

The only simple algebraic group *G* satisfying \(\lambda (G)=l(G)\) is \(G = A_1\).

### Proof

First observe that \(\lambda (A_1)=l(A_1)=3\), by Lemma 2.3. Conversely, suppose *G* is simple of rank \(r>1\) and \(\lambda (G)=l(G)\). We know that \(l(G) = l(B)+r\) by Corollary 2. If \(r\leqslant 4\) or *G* is exceptional, this contradicts Lemma 3.2 or Theorem 3.3. And if \(r\geqslant 5\), then Theorem 5(ii) gives a contradiction. \(\square \)

Now we prove Theorem 6. Let *G* be a connected algebraic group over an algebraically closed field. Suppose \(\lambda (G)=l(G)\) and *G* is insoluble. Set \(\bar{G} = G/R(G)\) and note that \(\lambda (\bar{G}) = l(\bar{G})\). Write \(\bar{G} = G_1 \cdots G_t\), where each \(G_i\) is simple. Then \(\lambda (G_i) = l(G_i)\) for each *i*. By Lemma 3.12, this implies that \(G_i \cong A_1\) for all *i*. Since \(\lambda (A_1A_1)=4 < l(A_1A_1)=6\), we must have \(t=1\), so \(\bar{G} \cong A_1\), as in Theorem 6.

### 3.8 Proof of Theorem 7

Let *G* be an algebraic group and recall that \(\mathrm{cd}(G) = l(G) - \lambda (G)\) is the chain difference of *G*.

### Lemma 3.13

*N*is a connected normal subgroup of

*G*, then

### Proof

By Lemma 2.1 we have \(l(G) = l(N) + l(G/N)\) and \(\lambda (G) \leqslant \lambda (N) + \lambda (G/N)\). The conclusion follows.

The analogous result for finite groups is [3, Lemma 1.3].

We now state some immediate consequences of the above lemma.

### Corollary 3.14

- (i)
If

*N*is a connected normal subgroup of*G*, then \(\mathrm{cd}(G/N) \leqslant \mathrm{cd}(G)\). - (ii)
If \(1 = G_t \lhd G_{t-1} \lhd \cdots \lhd G_1 \lhd G_0 = G\) is a chain of connected subgroups of

*G*, then \(\mathrm{cd}(G) \geqslant \sum _{i} \mathrm{cd}(G_{i-1}/G_i)\). - (iii)
If \(G = G_1 \times \cdots \times G_t\), where each \(G_i\) is connected, then \(\mathrm{cd}(G) \geqslant \sum _{i} \mathrm{cd}(G_i)\).

The next result bounds \(\dim G\) in terms of \(\mathrm{cd(G)}\) when *G* is simple.

### Proposition 3.15

*G*be a simple algebraic group in characteristic \(p \geqslant 0\). Then

### Proof

*r*be the rank of

*G*. Using Corollary 2 and its notation we obtain

*c*be the value of \(\lambda (G)\) as in Theorem 4. Then we have

The next result is of a similar flavour, dealing with certain semisimple groups.

### Proposition 3.16

*S*is a simple algebraic group in characteristic \(p \geqslant 0\). Then

### Proof

*B*is a Borel subgroup of

*S*, and \(r = \mathrm{rank}(S)\), then

If \(p=0\) then it is easy to check using Theorem 4 that \(a \leqslant 2\) in all cases, as required.

### Lemma 3.17

Let \(S_1,\ldots ,S_n\) be simple algebraic groups that are pairwise non-isomorphic. Then \(\sum _{i} \dim S_i \geqslant n^2\).

### Proof

*r*, the number of distinct types of simple algebraic groups of rank at most

*r*is at most 4

*r*. Hence \(\mathrm{rank}(S_i) \geqslant \frac{i}{4}\), and so

*G*be a connected algebraic group. If

*G*is soluble then the conclusion holds trivially, so suppose

*G*is insoluble. Let

*R*(

*G*) be the radical of

*G*and write

### Remark 3.18

Let *G* be an algebraic group in characteristic \(p \geqslant 0\) and set \(\bar{G}=G/R(G)\). For a simple group *G*, it is easy to see that \(\mathrm{cd}(G)=1\) if and only if \(G = A_2\) and \(p=2\). In the general case, by arguing as in the proof of Theorem 6, one can show that \(\mathrm{cd}(G)=1\) only if \(\bar{G} = A_1\), or \(p = 2\) and \(\bar{G} = A_2\). For example, if \(G = UA_1\), a semidirect product where *U* is the natural module for \(A_1\), then \(l(G)=5\) and \(\lambda (G)=4\).

## 4 Chain ratios

*chain ratio*\(\mathrm{cr}(G)\) of an algebraic group

*G*, which is defined by

*G*is simple, then its dimension is bounded in terms of its chain ratio.

### Proposition 4.1

*G*be a simple algebraic group in characteristic \(p \geqslant 0\). Then

- (i)
\(\dim G < 12\,\mathrm{cr}(G)\) if \(p=0\).

- (ii)
\(\dim G < (1+o(1)) \cdot \mathrm{cr}(G) \cdot {(\log _2 \mathrm{cr}(G))}^2\) if \(p > 0\), where \(o(1) = o_{\mathrm{cr}(G)}(1)\).

### Proof

Assuming *G* is simple and \(p=0\), we have \(\lambda (G) \leqslant 6\) by Theorem 4, proving part (i).

*r*be the rank of

*G*. Then \(\lambda (G) \leqslant 2(\log _2 r)^2 + 12\) by Theorem 5. Since \(d>r^2\) we obtain \(\lambda (G) < \frac{1}{2}(\log _2d)^2 + 12\), so

In contrast to this result, we shall exhibit a sequence of algebraic groups *G* for which \(\dim G/R(G)\) is not bounded above in terms of the chain ratio \(\mathrm{cr}(G)\). To show this we need the following result.

### Lemma 4.2

If *S* is a simple algebraic group and \(k \in \mathbb {N}\), then \(\lambda (S^k) \geqslant k+2\).

### Proof

The proof goes by induction on *k*, the case \(k=1\) being clear.

Suppose \(k>1\), write \(G=S^k\) and let \(\pi _i:G\rightarrow S\) be the projection to the *i*-th factor. Let *M* be a maximal connected subgroup of *G* with \(\lambda (M)=\lambda (G)-1\). If \(\pi _i(M)=M_i<S\) for some *i*, then \(M = M_i \times S^{k-1}\), and so \(\lambda (M) \geqslant \lambda (S^{k-1}) \geqslant k+1\) by induction, giving the conclusion.

Now assume \(\pi _i(M)=S\) for all *i*. Then *M* is a product of diagonal subgroups of various subsets of the simple factors of \(S^k\), and maximality forces \(M = \mathrm{diag}(S^2)\times S^{k-2}\), where \(\mathrm{diag}(S^2)\) denotes a diagonal subgroup of \(S^2\). Hence \(M \cong S^{k-1}\) and the conclusion again follows by induction. \(\square \)

*S*and let \(G = S^k\) for \(k \geqslant 1\). Since \(l(G) = k\cdot l(S)\) and \(\lambda (G) \geqslant k+2\) by Lemma 4.2, it follows that

*k*tend to infinity, we see that \(\mathrm{cr}(G)\) is bounded, while \(\dim G/R(G) = \dim G\) tends to infinity.

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