FINITE p-GROUPS WHICH CONTAIN A SELF-CENTRALIZING CYCLIC NORMAL SUBGROUP

For any prime p, all finite noncyclic p-groups which contain a self-centralizing cyclic normal subgroup are determined by using cohomological techniques. Some applications are given, including a character theoretic description for such groups.

The influence of X-semipermutability of subgroups on the structure of finite groups

Let A be a subgroup of a group G and X be a nonempty subset of G. A is said to be X-semipermutable in G if A has a supplement T in G such that A is X-permutable with every subgroup of T. In this paper, we investigate further the influence of X-semipermutability of some subgroups on the structure of finite groups. Some new criteria for a group G to be supersoluble or p-nilpotent are obtained.

Finite p-Groups in Which the Number of Subgroups of Possible Order Is Less Than or Equal to p^3

In this paper, groups of order p^n in which the number of subgroups of possible order is less than or equal to p3 ale classified. It turns out that if p 〉 2, n≥ 5, then the classification of groups of order p^n in which the number of subgroups of possible order is less than or equal to p3 and the classification of groups of order p^n with a cyclic subgroup of index p2 are the same.

Automorphisms of a Class of Finite p-groups with a Cyclic Derived Subgroup

Let p be an odd prime,and let k be a nonzero nature number.Suppose that nonabelian group G is a central extension as follows1→G’→G→Z_(pK)×…×Z_(pK),where G’≌Zpk,andζG/G’is a,direct factor of G/G’.Then G is a central product of an extraspecial pkgroup E andζG.Let|E|=p(2n+1)k and|ζG|=p(m+1)k.Suppose that the exponents of E andζG are pk+l and pk+r,respectively,where 0≤l,r≤k.Let AutG’G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G’,let AutG/ζG,ζG G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the centerζG and let AutG/ζG,ζG/G’G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially onζG/G’.Then(ⅰ)The group extension 1→Aut G’→Aut G→Aut G’→1 is split.(ⅱ)AutG’G/AutG/ζG,ζG G≌G1×G2,where Sp(2n-2,Zpk)■H≤G1≤Sp(2n,Zpk),H is an extraspecial pk-group of order p(2n-1)k and(GL(m-1,Zpk)■Zpk(m-1)■Zpk(m)≤G2≤GL(m,Zpk)■Zpk(m).In particular,G1=Sp(2n-2,Zpk)■H if and only if l=k and r=0;G1=Sp(2n,Zpx)if and only if l≤r;G2=(GL(m-1,Zpk)■Zpk(m-1)■Zpk(m)if and only if r=k;G2=GL(m,Zpk)■Zpk((m))if and only if r=0.(ⅲ)AutG’G/Aut G/ζG,ζG/G’G≌G1×G3,where G1 is defined in(ⅱ);GL(ml,Zpk)■Zpk(m-1)≤G3≤GL(n,Zpk).In particular,G3=GL(m-1,Zpk)■Zpk(m-1)if and only if r=k;G3=GL(m,Zpk)if and only if r=0.(ⅳ)AntG/ζG,ζG/G’G≌AutG/ζG,ζG/G’G■Zpk(m),If m=0,then AntG/ζG,ζG/G’G=Inn G≌Zpk(2n);If m>0,then AntG/ζG,ζG/G’G≌Zpk(2nm)×Zpk-r(2n),and AutG/ζG,ζG G/Inn G≌Zpk((2n(m-1))×Zpk-r(2n).

A Characteristic of Dedekind Groups

In this paper, we show that if a finite group G has a fixed-point-free weak power automorphism, then G is a Dedekind group.

Finite NPDM-groups

In this paper the classification is given for finite groups in which the normalizer of every non-normal cyclic subgroup of order divided by the minimal prime of |G| is a maximal subgroup.