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A Class of Finite 2-groups G with Every Automorphism Fixing \(G/\varPhi (G)\) Elementwise

  • Hossein AbdolzadehEmail author
  • Reza Sabzchi
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 228)

Abstract

The family \(G(m,n)=\langle x,y| x^2=(xy^2)^2=1,~y^{2^m}=(xy)^{2^{n}}\rangle \) of finite 2-groups will be introduced. The group G(mn) has order \(2^{(m+n+1)}\), nilpotency class \(1+\max \{m,n\}\) and every automorphism of \(G=G(m,n)\) fixes \(G/\varPhi (G)\) elementwise and therefore Aut(G) is a 2-group. The parameterized presentation of \(G=G(m,n)\) is efficient as the Schur multiplicator of G is non-trivial.

Keywords

Finite 2-group Automorphism group Frattini subgroup 

2010 MSC:

Primary 20D15 Secondary 20D45 20F05 

References

  1. 1.
    Adney, J.E., Yen, T.: Automorphisms of a \(p\)-group. Illinois J. Math. 9, 137–143 (1965)Google Scholar
  2. 2.
    Caranti, A., Scoppola, C.M.: Endomorphisms of two-generated metabelian groups that induce the identity modulo the derived subgroup. I. Arch. Math. 56(3), 218–227 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Liebeck, H.: The automorphism group of finite p-groups. J. Algebra 4, 426–432 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Jafari, M.H.: Elementary abelian \(p\)-groups as central automorphism groups. Comm. Algebra 34, 601–607 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Jamali, A.R.: \(2\)-groups \(G\) with every automorphism fixing \(G/\varPhi (G)\) elementwise. Southeast Asian Bull. Math. 31, 255–257 (2007)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Johnson, D. L.: Presentations of Groups. London Math. Soc. Stud. Texts. Cambridge University Press, Cambridge 15 (1990)Google Scholar
  7. 7.
    Karpilovsky, G.: The Schur Multiplier. Clarendon Press, Oxford (1987)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Applications, Faculty of SciencesUniversity of Mohaghegh ArdabiliArdabilIran

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