Stewart Stonehewer

• MA (Oxon), PhD (Cantab)
• Emeritus Professor at University of Warwick

Quasinormal subgroups have been studied for nearly 80 years. In finite groups, questions concerning them invariably reduce to p-groups, and here they have the added interest of being invariant under projectivities, unlike normal subgroups. However, it has been shown recently that certain groups, constructed by Berger and Gross in 1982, of an import...

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 expo...

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.

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...

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...

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 p 2 in a finite p-group G contains a quasinorm...

Without Abstract

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 w...

A qn G: 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...

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 $A$. AMS 2000 Mat...

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...

In earlier work, the authors described the structure of the normal closure of a cyclic permutable subgroup of odd order in a finite group. As might be expected, the even order case is considerably more complicated and we have found it necessary to divide it into two parts. This part deals with the situation where we have a finite group $G$ with a c...

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.

A dinilpotent group is a group that can be written as the product of two nilpotent subgroups. There is an extensive literature dealing with such groups (see, for example, the recent book of Amberg, Franciosi and de Giovanni [1]). In 1955, Itô proved in [6] that the product of two abelian groups is always metabelian, and in the following year, Hall...

The existence of Tarski groups and other infinite groups with restricted subgroup lattices show that there are non-soluble groups with a duality. The present work finds restrictions imposed on an arbitrary group by the existence of a duality and studies some particular kinds of dualities.

The existence of Tarski groups and other infinite groups with restricted subgroup lattices show that there are non-soluble groups with a duality. The present work finds restrictions imposed on an arbitrary group by the existence of a duality and studies some particular kinds of dualities.

A survey is given of recent results on the structure of a group G which has a factorization of the form G=AB=BC=CA where A, B, C are Abelian subgroups.

A subgroup D of a group G is called dual-standard if, for all subgroups X and Y of G, [X intersects D, Y intersects D] = [X, Y] intersects D. When G is finite, Zappa has given some information concerning the way in which D is embedded in G and the structure of G itself. Among other things, Zappa makes reference to the maximal normal Hall subgroup L...

Let G be a group and n (≧ 2) an integer. We say that G belongs to the class of groups P n if every product of n elements can be reordered, i.e. for all n -tuples , there exists a non-trivial element σ in the symmetric group Σ n such that Let P denote the union of the classes P n , n ≧ 2. Clearly every finite group belongs to P and each class P n is...

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 subgroup H of a group G is said to be a permutable subgroup 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...

In (3) W. Gaschütz introduced the concept of a formation of finite soluble groups, generalizing the results of R. W. Carter on the existence and conjugacy of nilpotent self-normalizing subgroups in finite soluble groups (2). Carter's results have already been extended to two classes of infinite groups in (9), (10), and the object of the present wor...