Integrals of groups

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

Research output: Contribution to journalArticlepeer-review


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.
Original languageEnglish
Pages (from-to)149-178
JournalIsrael Journal of Mathematics
Issue number1
Early online date9 Sept 2019
Publication statusPublished - Oct 2019


  • Groups
  • Derived group
  • Inverse problem


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