Integrals of groups

Autor: Carlo Casolo, Peter J. Cameron, Francesco Matucci, João Araújo
Přispěvatelé: Araujo, J, Cameron, P, Casolo, C, Matucci, F, University of St Andrews. Pure Mathematics, University of St Andrews. Centre for Interdisciplinary Research in Computational Algebra
Jazyk: angličtina
Rok vydání: 2019
Předmět:
Popis: An $integral$ of a group $G$ is a group $H$ whose derived group (commutator subgroup) is isomorphic to $G$. This paper discusses integrals of groups, and in particular questions about which groups have integrals and how big or small those integrals can be. Our main results are: (1) If a finite group has an integral, then it has a finite integral. (2) A precise characterization of the set of natural numbers $n$ for which every group of order $n$ is integrable: these are the cubefree numbers $n$ which do not have prime divisors $p$ and $q$ with $q\mid p-1$. (3) An abelian group of order $n$ has an integral of order at most $n^{1+o(1)}$, but may fail to have an integral of order bounded by $cn$ for constant $c$. (4) A finite group can be integrated $n$ times (in the class of finite groups) if and only if it is the central product of an abelian group and a perfect group. There are many other results on such topics as centreless groups, groups with composition length $2$, and infinite groups. We also include a number of open problems.
31 pages, no figures; new co-author and new title; the previous posting has been split in half, with the second part to be expanded and resubmitted separately
Databáze: OpenAIRE
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