Minimal verbal subgroups 349

THEOREM 2. If G is a finite monolithic group with monolith M (in other words, if G is a finite group with a unique minimal normal subgroup M), then Mis a verbal subgroup ofG. These generalize the corresponding results for finite ^-groups established by Weichsel ((4), Theorems 2-1, 2-3). The second answers affirmatively a (privately communicated) question of Sheila Oates. In order to give our description we need some notation. Verbal subgroups can be defined in a number of ways. For our purposes the following procedure seems the most efficient. A variety of groups is a class of groups closed under forming subgroups, homomorphic images, and unrestricted direct products. Given a class C of groups there is a smallest variety containing it; this will be denoted var C or var G if C consists of the one group G. Let V be a variety, then in every group G there is a unique normal subgroup ^minimal with respect to the property that the factor group GjN lies in V. This normal subgroup is the verbal subgroup of G corresponding to the variety V and is denoted V(C?). Clearly \(G) = E (the identity subgroup) if and only if G € V. A minimal verbal subgroup of a group G is a non-trivial verbal subgroup of G which does not properly contain a non-trivial verbal subgroup of G. Let S be a subgroup of a group G. The centralizer of S in G, that is,