Groups that have a partition by commuting subsets

Let 𝐺 be a nonabelian group. We say that 𝐺 has an abelian partition if there exists a partition of 𝐺 into commuting subsets A_{1},A_{2},\ldots,A_{n} of 𝐺 such that \lvert A_{i}\rvert\geqslant 2 for each i=1,2,\ldots,n. This paper investigates problems relating to groups with abelian partitions. Among other results, we show that every finite group is isomorphic to a subgroup of a group with an abelian partition and also isomorphic to a subgroup of a group with no abelian partition. We also find bounds for the minimum number of partitions for several families of groups which admit abelian partitions – with exact calculations in some cases. Finally, we examine how the size of a partition with the minimum number of parts behaves with respect to the direct product.

Automatic realization of Hopf Galois structures

We consider Hopf Galois structures on a separable field extension [Formula: see text] of degree [Formula: see text], for [Formula: see text] an odd prime number, [Formula: see text]. For [Formula: see text], we prove that [Formula: see text] has at most one abelian type of Hopf Galois structures. For a nonabelian group [Formula: see text] of order [Formula: see text], with commutator subgroup of order [Formula: see text], we prove that if [Formula: see text] has a Hopf Galois structure of type [Formula: see text], then it has a Hopf Galois structure of type [Formula: see text], where [Formula: see text] is an abelian group of order [Formula: see text] and having the same number of elements of order [Formula: see text] as [Formula: see text], for [Formula: see text].

Cohomological Group Theory

This chapter introduces some of the basic ingredients of cohomological group theory and describes datatypes and algorithms for implementing them on a computer. These are illustrated using computer examples involving: explicit cocycles, classification of abelian and nonabelian group extensions, crossed modules, crossed extensions, five-term exact sequences, Hopf’s formula, Bogomolov multipliers, relative central extensions, nonabelian tensor products of groups, and cocyclic Hadamard matrices.

The largest conjugacy class size and the nilpotent subgroups of finite groups

LetHbe a nilpotent subgroup of a finite nonabelian groupG, let{\pi=\pi(|H|)}and let{{\operatorname{bcl}}(G)}be the largest conjugacy class size of the groupG. In the present paper, we show that{|HO_{\pi}(G)/O_{\pi}(G)|<{\operatorname{bcl}}(G)}.

Some New Groups which are not CI-groups with Respect to Graphs

A group $G$ is a CI-group with respect to graphs if two Cayley graphs of $G$ are isomorphic if and only if they are isomorphic by a group automorphism of $G$. We show that an infinite family of groups which include $D_n\times F_{3p}$ are not CI-groups with respect to graphs, where $p$ is prime, $n\not = 10$ is relatively prime to $3p$, $D_n$ is the dihedral group of order $n$, and $F_{3p}$ is the nonabelian group of order $3p$.

A nilpotency criterion related to the minimal degree of a nonlinear irreducible character

Let [Formula: see text] be a finite nonabelian group, let [Formula: see text] be the minimal degree of a nonlinear irreducible character of [Formula: see text] and suppose that [Formula: see text] for some positive integer [Formula: see text]. Then [Formula: see text] is nilpotent.