2003/02/11 by Maria S. Voloshina, Voloshina, Maria S. · 1 citation
Mathematics · #18G05 #18G40 (secondary) #20E22 #20G10 #20G40 #20J06 (primary) #20J15 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #math.AT #math.GR #msc:18G05 #msc:18G40 #msc:20E22 #msc:20G10 #msc:20G40 #msc:20J06 #msc:20J15
paper · pdf · doi:10.48550/arxiv.math/0302120
87 pages, 12 figures. Ph.D. dissertation, University of Rochester. Advisor: Frederick R. Cohen
openalex publication_date 2003/02/11 · arxiv created 2004/01/14 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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(Zpr) for every prime p, where \mathbb Zm denotes a cyclic group of order m, and use it to compute the integer homology and mod p cohomology ring of Hol(Zpr). In chapter 4 we study the holomorph of the direct sum of several copies of Zpr. We identify this holomorph as a nice subgroup of GL(n+1, Zpr), 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(\bigoplusn Zpr); Fp) does not collapse at the E2 stage for pr≥ 8. Also, we compute mod p cohomology and the first Bockstein homomorphisms of the congruence subgroups given by Ker (Hol(\bigoplusn Zpr) → Hol(\bigoplusn Zp)). 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.