On the holomorph of a discrete group

TL;DR

This paper explores the structure, properties, and cohomology of holomorphs of discrete groups, providing new resolutions, computations, and insights into their algebraic and topological characteristics.

Contribution

It introduces a new resolution for holomorphs of cyclic groups, computes their cohomology, and analyzes the cohomological behavior of holomorphs of direct sums, advancing understanding of their algebraic structure.

Findings

Constructed a resolution for $Hol(Z_{p^r})$ and computed its homology and cohomology.

Identified holomorphs as subgroups of $GL(n+1, Z_{p^r})$ and analyzed their cohomology.

Showed the LHS spectral sequence does not collapse at $E_2$ for certain holomorphs.

Abstract

The holomorph of a discrete group $G$ is the universal semi-direct product of $G$. In chapter 1 we describe why it is an interesting object and state main results. In chapter 2 we recall the classical definition of the holomorph as well as this universal property, and give some group theoretic properties and examples of holomorphs. In particular, we give a necessary and sufficient condition for the existence of a map of split extensions for holomorphs of two groups. In chapter 3 we construct a resolution for $Hol(Z_{p^r})$ for every prime $p$, where ${\mathbb Z}_m$ denotes a cyclic group of order $m$, and use it to compute the integer homology and mod $p$ cohomology ring of $Hol(Z_{p^r})$. In chapter 4 we study the holomorph of the direct sum of several copies of $Z_{p^r}$. We identify this holomorph as a nice subgroup of $GL(n+1, Z_{p^r})$, thus its cohomology informs on the cohomology of the general linear group which has been of interest in the subject. We show that the LHS spectral sequence for $H^*(Hol(\bigoplus_n Z_{p^r}); F_p)$ does not collapse at the $E_2$ stage for $p^r\ge 8$. Also, we compute mod $p$ cohomology and the first Bockstein homomorphisms of the congruence subgroups given by $Ker (Hol(\bigoplus_n Z_{p^r}) \to Hol(\bigoplus_n Z_p)).$ In chapter 5 we recall wreath products and permutative categories, and their connections with holomorphs. In chapter 6 we give a short proof of the well-known fact due to S. Eilenberg and J. C. Moore that the only injective object in the category of groups is the trivial group.