On the number of commutators in groups

By commutator in a group G is meant the derived (or commutator) subgroup of a subgroup of G. It is a natural question how important the set of commutator subgroups is within the lattice of all subgroups. In this paper we are interested in the effect of imposing restrictions on the number of derived subgroups in a group and investigating the resulting effect on the structure of the group. Let C_n denote the class of groups in which there are at most n+1 derived subgroups, and let C denote the union of all the classes C_n, n = 0, 1, 2, . . . . Obviously C_0 is exactly the class of the abelian groups. The class C_1 is much more interesting. It contains, clearly, all groups satisfying |G′| = p for some prime p, but also the class of groups, called the Tarski Monsters, infinite simple groups, with all proper non-trivial subgroups having a fixed prime order p. Such groups were shown to exist for primes p large enough (p > 1075) by Rips and Ol’shankiˇi. In this paper we study the classes C_n for small n. We also study groups in C. We show, among other things, that a group G in C must have its commutator subgroup finitely generated. Moreover, a group G has a finite commutator subgroup if and only if G is a locally graded group in C.