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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.

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 location of quasinormal subgroups in a group is not particularly well known. Maximal ones always have to be normal, but little has been proved about the minimal ones. In finite groups, the difficulties arise in the p-groups. Here we prove that, for every odd prime p, a quasinormal subgroup of order p2 in a finite p-group G contains a quasinormal subgroup of G of order p.

Thus normal subgroups are always quasinormal, but not conversely. For, if p is a prime, then any cyclic group Cpn extended by any cyclic group Cpm has all subgroups quasinormal (provided, when p = 2 and n > 2, the cyclic subgroup of order 4 in C2n is central in the extension). The same is true if Cpn is replaced by any abelian p-group H of finite exponent, with Cpm acting on H as a group of power automorphisms (and elements of order 4 in H are again central in the extension if p = 2). These results can be found in sections 2.3 and 2.4 of [16]. One of the earliest results about quasinormal subgroups is due to Ore, who proved in 1938 that a quasinormal subgroup of a finite group G is always subnormal in G ([14]). When G is infinite, then A does not have to be subnormal, but it is always ascendant in G (see [17]). Clearly the extent to which a quasinormal subgroup A can differ from being normal is of interest and a measure of this was given by Ito and Szep in 1962 when they proved that, again with G finite, and denoting the core of A in G by AG, the quotient A/AG is always nilpotent

The authors obtain a bound for the derived length of a finite group, which is the join of two disjoint permutable nilpotent subgroups, in terms of the derived lengths of the two subgroups.

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

If a finite group G is the product of two nilpotent subgroups A and B and if N is a minimal normal subgroup of G , then AN or BN is nilpotent. This result is extended to several classes of infinite groups.

In Cossey and Stonehewer [‘On the rarity of quasinormal subgroups’, Rend. Semin. Mat. Univ. Padova125 (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 pn containing a quasinormal subgroup H of exponent pn−1 such that the nontrivial quasinormal subgroups of G lying in H can have exponent only p, pn−1 or, when n≥4 , pn−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.

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.

Usually we shall write simply t: C I + Y to denote the map r and speak of a complete /-homomorphism from G to 2’. We call T trivial if all subgroups of G have the same image under s; and we call T proper if r is not trivial and not injective. If (t) holds for all finite subsets .(/‘. then 7 is called a Irrtticr /2o111ot?1orpJli.o}?l (or I-i~~~~onlorpllisnz ). An /-homomorphism from G to the lattice /(G) of a group C? is called a projectiuit~~ if it is a bijection. Of course, a projectivity is always complete. A projectivity a: G --f G is said to be ilz~k~.~-prc.vercitl~ if. for K < H < G with 1 H : K1 finite.

The main purpose of this paper is to exhibit a doubly-infinite family of examples which are extensions of a p-group by a p′-group, with the action satisfying some conditions of Zappa (1951), arising from his study of dual-standard (meet-distributive) subgroups. The examples show that Zappa's conditions do not bound the nilpotency class (or even the derived length) of the p-group. The key to this work is found in closely related conditions of Hartley (published here for the first time). The examples use some exceptional relationships between primes.

Let G be a group with a core-free subgroup H (≠1) such that the interval [ G / H ] is a projective geometry. Then H has a normal abelian complement (in G ) on which H acts faithfully and which is the direct product of H -isomorphic minimal normal subgroups of G provided either G is hyperabelian and H possesses a finite cyclic normal subgroup ≠1 (Theorem A) or G has an ascending normal series with finite abelian factors (Theorem B).

The Jacobson radicals of group algebras have received the attention of many authors over the years and several problems have remained unsolved. For example (see [#] and [.5]): If G is a group a& if F is a field of characteristic 0, is the group algebra FG semi-sinzple? (FG is semi-simple if its Jacobson radical J(FG) is triviahj It is known that FG is semi-simple in various special cases; for example, if

A subgroup H of a group G is said to be a permutablesubgroup of G if HK = KH 〈H, K〉 for all subgroups K of G. It is known that a core-free permutable subgroup H of a finite group G is always nilpotent [5]; and even when G is not finite, H is always a subdirect product of finite nilpotent groups [11]. Thus nilpotency is a measure of the extent to which a permutable subgroup differs from being normal. Examples of non-abelian, core-free, permutable subgroups are rare and difficult to construct. The first, due to Thompson [12], had class 2. Further examples of class 2 appeared in [8]. More recently Bradway, Gross and Scott [1] have constructed corresponding to each positive integer c and each prime p ≥ c, a finite p-group possessing a core-free permutable subgroup of class c. In [3] Gross succeeded in dispensing with the requirement p ≥ c.