The Structure And Its Automorphism Group Of A Class Of Finite 2-Groups

Prom the aspect of involutions,finite 2-groups with a maximal subgroup isomor-phic to a generalized quaternion group are studied.By investigating the conjugacy classes of involution automorphisms and the number of involutions in the correspond-ing semi-direct products,a complete classification of such 2-group is obtained.In particular,the number of involutions in such 2-groups is shown to be a completely invariant under isomorphisms.Furthermore,from the viewpoint of involutions,the structure of automorphism groups of modular 2-groups is given.The main conclusions of this paper are as follows:Theorem 1.Let G be a finite 2-group such that there is a maximal subgroup iso-morphic to a generalized quaternion group Q2n+1,where Q2n+1:<a,b | a2n = 1,b2 =a2n-1 ab ?a-1>,n ? 3.Then G is isomorphic to exactly one of the following groups:(1)Q2n+2.(2)Q2n+1× C2;(3)Q2n+1 ×C2,group actions:ac=a-1,bc = b;(4)Q2n+1× C2,group actions:ac = a-1,bc = ba;(5)Q2n+1× C2,group actions:ac = a1+2n-1,bc = b;where C2 =<c>is a cyclic group of order 2.In the proof of Theorem 1,we find that the number of involutions is a completely invariant under isomorphism.Theorem 2.Let G and H are finite 2-groups and both have a max,imal subgroup isomorphic to a generalized quaternion group Q2n+1,n ? 3.Then G ? H if and only if G and H have the same number of involutions.Finally,we give a description of the automorphism groups of modular 2-groups.Theorem 3.Let M2n+1 = Ma,b | a2n = 1,b2 = 1,ab = a1+2n-1>is a modular 2-group,n ? 3.ThenAut(M2n+1)=<?>×((?>×(<?-1>×<?5>))? C2×(C2×U(Z2n).