A criterion for the existence of nilpotent hall π-subgroups

Let [Formula: see text] be a nontrivial normal subgroup of [Formula: see text] and [Formula: see text]. In this paper, we show that if [Formula: see text] and for every [Formula: see text]-element [Formula: see text] with [Formula: see text], [Formula: see text], for some integer [Formula: see text], then [Formula: see text] has nilpotent Hall [Formula: see text]-subgroups. Further, we show that if [Formula: see text], then [Formula: see text] has abelian Hall [Formula: see text]-subgroups.

Free Groups and Folding

This chapter studies subgroups of free groups using the combinatorics of graphs and a simple operation called folding. It introduces a topological model for free groups and uses this model to show the rank of the free group H and whether every finitely generated nontrivial normal subgroup of a free group has finite index. The edge paths and the fundamental group of a graph are discussed, along with subgroups via graphs. The chapter also considers five applications of folding: the Nielsen–Schreier Subgroup theorem, the membership problem, index, normality, and residual finiteness. A group G is residually finite if for every nontrivial element g of G there is a normal subgroup N of finite index in G so that g is not in N. Exercises and research projects are included.

THE NUMBER OF PROFINITE GROUPS WITH A SPECIFIED SYLOW SUBGROUP

Let $S$ be a finitely generated pro-$p$ group. Let ${\mathcal{E}}_{p^{\prime }}(S)$ be the class of profinite groups $G$ that have $S$ as a Sylow subgroup, and such that $S$ intersects nontrivially with every nontrivial normal subgroup of $G$. In this paper, we investigate whether or not there is a bound on $|G:S|$ for $G\in {\mathcal{E}}_{p^{\prime }}(S)$. For instance, we give an example where ${\mathcal{E}}_{p^{\prime }}(S)$ contains an infinite ascending chain of soluble groups, and on the other hand show that $|G:S|$ is bounded in the case where $S$ is just infinite.

Bounds on finite quasiprimitive permutation groups

A permutation group is said to be quasiprimitive if every nontrivial normal subgroup is transitive. Every primitive permutation group is quasiprimitive, but the converse is not true. In this paper we start a project whose goal is to check which of the classical results on finite primitive permutation groups also holds for quasiprimitive ones (possibly with some modifications). The main topics addressed here are bounds on order, minimum degree and base size, as well as groups containing special p-elements. We also pose some problems for further research.

ULC Properties in Neighbourhoods of Embedded Surfaces and Curves in E3

In this paper we derive those properties of topologically embedded curves and surfaces in E3 which can be obtained without use of Bing's Side Approximation Theorem [3] for surfaces. The local homology and homotopy properties studied classically play the largest role in the paper, but the final geometrization of some of the results requires theorems such as the PL Schoenflies Theorem, Dehn's Lemma, the Loop Theorem, the Sphere Theorem, and Waldhausen's generalization of the Loop Theorem (n.b., one application of Waldhausen's theorem (in (3.4)) requires use of the nontrivial normal subgroup in the statement of that theorem).

Finite groups with a self-centralizing subgroup of order 4

The class of finite groups having a subgroup of order 4 which is its own centralizer has been studied by Suzuki [9], Gorenstein and Walter [6], and the present author [11]. The main purpose of this paper is to strengthen Theorem 5 of [11] by using an early result of Zassenhaus [12]. In particular, we find all groups of the class which are core-free, i.e. which have no nontrivial normal subgroup of odd order. As an application, we make a determination of a certain class of primitive permutation groups.