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: (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.