On a set of normal subgroups

Abstract

The commutator [a, b] of two elements a and b in a group G satisfies the identityab = ba[a, b].The subgroups we study are contained in the commutator subgroup G′, which is the subgroup generated by all the commutators.The group G is covered by a well-known set of normal subgroups, namely the normal closures {g}G of the cyclic subgroups {g} in G. In a similar way one may associate a subgroup K(g) with each element g, by defining K(g) to be the subgroup generated by the commutators [g, x] as x takes all values in G. These subgroups generate G′ (but do not cover G′ in general), and are normal in G in consequence of the identical relation(A) [g, x]Y = [g, y]−1[g, xy]holding for all g, x and y in G. (By ab we mean b−1ab.) It is easy to see that{g}G = {g, K(g)}.