On the holomorph of a discrete group
Abstract
Description
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.
87 pages, 12 figures. Ph.D. dissertation, University of Rochester. Advisor: Frederick R. Cohen
87 pages, 12 figures. Ph.D. dissertation, University of Rochester. Advisor: Frederick R. Cohen