Integrals of groups

João Araújo; Peter Jephson Cameron; Carlo Casolo; Francesco Matucci

doi:10.1007/s11856-019-1926-y

Integrals of groups

João Araújo, Peter Jephson Cameron, Carlo Casolo, Francesco Matucci

Research output: Contribution to journal › Article › peer-review

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Abstract

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:

- If a finite group has an integral, then it has a finite integral.
- 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 | p-1.
- 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.
- A finite group can be integrated n times (in the class of finite groups) for every n if and only if it is a 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.

Original language	English
Pages (from-to)	149-178
Journal	Israel Journal of Mathematics
Volume	234
Issue number	1
Early online date	9 Sept 2019
DOIs	https://doi.org/10.1007/s11856-019-1926-y
Publication status	Published - Oct 2019

Keywords

Groups
Derived group
Inverse problem

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Integrals_revised
Copyright © 2019 The Hebrew University of Jerusalem. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1007/s11856-019-1926-y
Accepted author manuscript, 394 KBLicence: Unspecified

Cite this

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title = "Integrals of groups",

abstract = "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:- If a finite group has an integral, then it has a finite integral.- 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 | p-1.- An abelian group of order n has an integral of order at most n1+o(1), but may fail to have an integral of order bounded by cn for constant c.- A finite group can be integrated n times (in the class of finite groups) for every n if and only if it is a 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.",

keywords = "Groups, Derived group, Inverse problem",

author = "Jo{\~a}o Ara{\'u}jo and Cameron, {Peter Jephson} and Carlo Casolo and Francesco Matucci",

note = "Funding: Funda{\c c}{\~a}o para a Ci{\^e}ncia e a Tecnologia (CEMAT-Ci{\^e}ncias FCT project UID/Multi/04621/2013).",

year = "2019",

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doi = "10.1007/s11856-019-1926-y",

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pages = "149--178",

journal = "Israel Journal of Mathematics",

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T1 - Integrals of groups

AU - Araújo, João

AU - Cameron, Peter Jephson

AU - Casolo, Carlo

AU - Matucci, Francesco

N1 - Funding: Fundação para a Ciência e a Tecnologia (CEMAT-Ciências FCT project UID/Multi/04621/2013).

PY - 2019/10

Y1 - 2019/10

N2 - 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:- If a finite group has an integral, then it has a finite integral.- 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 | p-1.- An abelian group of order n has an integral of order at most n1+o(1), but may fail to have an integral of order bounded by cn for constant c.- A finite group can be integrated n times (in the class of finite groups) for every n if and only if it is a 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.

AB - 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:- If a finite group has an integral, then it has a finite integral.- 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 | p-1.- An abelian group of order n has an integral of order at most n1+o(1), but may fail to have an integral of order bounded by cn for constant c.- A finite group can be integrated n times (in the class of finite groups) for every n if and only if it is a 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.

KW - Groups

KW - Derived group

KW - Inverse problem

U2 - 10.1007/s11856-019-1926-y

DO - 10.1007/s11856-019-1926-y

M3 - Article

SN - 0021-2172

VL - 234

SP - 149

EP - 178

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 1

ER -

Integrals of groups

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Keywords

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