The embedding of a cyclic permutable subgroup in a finite group. II

In two previous papers we established the structure of the normal closure of a cyclic permutable subgroup $A$ of a finite group, first when $A$ has odd order and second when $A$ has even order, but with an extra hypothesis that was unnecessary in the odd case. Here we describe the most general situation without any restrictions on $A$

Cyclic permutable subgroups of finite groups

The authors describe the structure of the normal closure of a cyclic permutable subgroup of odd order in a finite group

On the rarity of quasinormal subgroups

For each prime p and positive integer n, Berger and Gross have defined a finite p-group G = HX, where H is a core-free quasinormal subgroup of exponent p(n-1) and X is a cyclic subgroup of order p(n). These groups are universal in the sense that any other finite p-group, with a similar factorisation into subgroups with the same properties, embeds in G. In our search for quasinormal subgroups of finite p-groups, we have discovered that these groups G have remarkably few of them. Indeed when p is odd, those lying in H can have exponent only p, p(n-2) or p(n-1). Those of exponent p are nested and they all lie in each of those of exponent p(n-2) and p(n-1)

The subgroup lattice index problem

Given a group G and subgroups X >= Y, with Y of finite index in X, then in general it is not possible to determine the index |X : Y| simply from the lattice l(G) of subgroups of G. For example, this is the case when G has prime order. The purpose of this work is twofold. First we show that in any group, if the indices |X : Y| are determined for all cyclic subgroups X, then they are determined for all subgroups X. Second we show that if G is a group with an ascending normal series with factors locally finite or abelian, and if the Hirsch length of G is at least 3, then all indices |X : Y| are determined

Cyclic quasinormal subgroups of arbitrary groups

In recent years several papers have appeared showing how cyclic quasinormal subgroups are embedded in finite groups and many structure theorems have been proved. The purpose of the present work is twofold. First we show that, without exception, all of these theorems remain valid for finite cyclic quasinormal subgroups of infinite groups. Secondly we obtain analogous results for infinite cyclic quasinormal subgroups, where the statements turn out to be even stronger