The Genus of a Nilpotent R-Powered Group
| dc.contributor.author | Ginster, Carl | |
| dc.date.accessioned | 2023-11-10T10:29:26Z | |
| dc.date.available | 2023-11-10T10:29:26Z | |
| dc.date.issued | 2022 | |
| dc.description | A dissertation submitted in fulfilment of the requirements for the degree of Master of Science to the Faculty of Science, University of the Witwatersrand, Johannesburg, 2022 | |
| dc.description.abstract | In [14], Mislin define the genus G(N) of a finitely generated nilpotent group N to be the set of isomorphism classes of finitely generated nilpotent groups M such that the localizations Mp and Np are isomorphic at every prime p. In [6], Hilton and Mislin define an abelian group structure on the genus set G(N) of a finitely generated nilpotent group N with finite commutator subgroups. Let χ0 be the class of finitely generated groups with finite commutator subgroup. For a χ0-group G, the non cancellation set of G, denoted by χ(G), is the set of isomorphism classes of groups H such that G×Z ∼= H ×Z. Warfield, in [19], proved that, if N is a nilpotent χo-group, then G(N) = χ(N). In [20], the author showed that, for a χo-group G the non- cancellation set χ(G) has a group structure similar to the group structure on the Mislin genus of a nilpotent χo-group. Let R be a binomial ring. The nilpotent R-powered group, first introduced by P. Hall in [5], is a nilpotent group G extended by a binomial ring R. Many results that are found in the theory of nilpotent groups carry over to the class of nilpotent R-powered groups. In particular, Majewicz and Zyman in [13], showed that the P-localization of a nilpotent R-powered group G, for a set of primes P in R can be obtained. We study the genus of a finitely R-generated nilpotent R-powered group. We show that the for any two finitely R-generated nilpotent R-powered groups G and H and some finitely R-powered abelian group A, if G(G × A) = G(H × A) then we have G(G) = G(H) | |
| dc.description.librarian | PC(2023) | |
| dc.faculty | Faculty of Science | |
| dc.identifier.uri | https://hdl.handle.net/10539/36955 | |
| dc.language.iso | en | |
| dc.school | Mathematics | |
| dc.subject | Nilpotent R-Powered Group | |
| dc.subject | Genus G(N) | |
| dc.title | The Genus of a Nilpotent R-Powered Group | |
| dc.type | Dissertation |