Integrals of groups

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