Generalising quasinormal subgroups

In Cossey and Stonehewer ['On the rarity of quasinormal subgroups', Rend. Semin. Mat. Univ. Padova 125 (2011), 81-105] it is shown that for any odd prime p and integer n >= 3, there is a finite p-group G of exponent p(n) containing a quasinormal subgroup H of exponent p(n-1) such that the nontrivial quasinormal subgroups of G lying in H can have exponent only p, p(n-1) or, when n >= 4, p(n-2). Thus large sections of these groups are devoid of quasinormal subgroups. The authors ask in that paper if there is a nontrivial subgroup-theoretic property X: of finite p-groups such that (i) X is invariant under subgroup lattice isomorphisms and (ii) every chain of X-subgroups of a finite p-group can be refined to a composition series of X-subgroups. Failing this, can such a chain always be refined to a series of X-subgroups in which the intervals between adjacent terms are restricted in some significant way? The present work embarks upon this quest

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

Generalized Quasinormal Subgroups of Order p2p^2

In a recent paper, Cossey and Stonehewer introduced a generalization of the concept of quasinormal subgroups as a consequence of their discovery of a scarcity of the latter in certain finite $p$-groups of an important universal nature. Cyclic subgroups in this generalized class were shown to possess certain interesting properties, including their invariance under index-preserving projectivities. Naturally the first step in that work was the consideration of the subgroups of prime order. Thus, moving on from cyclic groups, a study of the non-cyclic subgroups of order $p^2$ in this generalized class would seem to be appropriate. It is shown here that they also are invariant under index-preserving projectivities

Abelian quasinormal subgroups of finite p-groups

If G = AX is a finite p-group, with A an abelian quasinormal subgroup and X a cyclic subgroup, then we find two composition series of G passing through A, all the members of which are quasinormal subgroups of G. (C) 20W Elsevier Inc