Finite 2-groups of Class 2 with Specific Automorphism Group

TL;DR

This paper classifies finite 2-groups of class 2 where all automorphisms of order 2 fixing the Frattini subgroup are inner, identifying a specific family of groups with this property.

Contribution

It provides a complete classification of such 2-groups, introducing the family Q(n; r) characterized by explicit relations and automorphism properties.

Findings

Groups Q(n; r) are characterized by specific relations involving n and r.

Automorphisms of order 2 fixing the Frattini subgroup are inner for these groups.

The classification covers all finite 2-groups of class 2 with this automorphism property.

Abstract

In this paper we classify all finite 2-groups of class 2 for which every automorphism of order 2 leaving the Frattini subgroup elementwise fixed is inner. We prove that every such group G is isomorphic to Q(n; r) = <a, b| a^{2n}= b^{2r}= 1; a^2^{n-r}= [a, b]> for some positive integers r; n such that 2 < 2r <= n; and every automorphism of Q(n; r) of order 2 leaving the Frattini subgroup elementwise fixed is inner.