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)


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.


Finite 2-group Automorphism group Frattini subgroup 

2010 MSC:

Primary 20D15 Secondary 20D45 20F05 


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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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