On soluble groups with finite homomorphic images of bounded rank

Abstract

Let G be a group. We denote by 9r(G) the set of all finite homomorphic images of G. The study of the connection between the properties of G and of groups of the set ~'(G) is a traditional problem of group theory. In this connection the following question, proposed by Wilson (see [1, Question 12.90]), is of interest: is a finitely generated residually finite group G minimax if there exists a finitely generated soluble minimax group H such that 9V(G) = 9V(H)? In the present paper we show that G is also minimax if ~-(G) C_ ~'(H). We recall that a group is minimax if it has a finite subnormal series each factor of which satisfies either the minimality condition or the maximality condition for subgroups. The basic properties of minimax groups are discussed in [2]. Lemma 1. Let G be a residually finite group, and let H be a group of finite rank. If ~(G) C ~(H), then G does not contain infinite elementary Abelian p-subgroups. Proof. Suppose that G contains an infinite elementary Abelian p-subgroup A. Then G contains a finite elementary Abelian p-subgroup B such that [B I > pr, where r is the rank of H. Since G is a residually finite group, we see that G contains a normal subgroup R of finite index such that B fq R = 1. Then G = G/R contains an elementary Abelian p-subgroup/~ such that IBI > p~. Since G E 37(G) and ~-(G) C ~-(g), we have G E 5r(g). Therefore, r(G) pL The lemma has been proved. Lemma 2. Let G be a subgroup of the Cartesian product G = :~iGi of finite soluble groups Gi of bounded orders. Suppose that H is a group of finite rank. If ~(G) C 3U(H), then G is a finite group. Proof. Since the orders of Gi are bounded, it follows that the solubility lengths of Gi are also bounded and, hence, G is soluble. Suppose that G is an infinite group and show by induction on the solubility length of G that the latter contains an infinite elementary Abelian p-subgroup. Assume that G is an Abelian group. Since there exists n E N such that IGil < n for all i, it follows that there exists t E N such that G~ = 1 for all i; therefore, G t = 1. From this it follows that G contains an infinite elementary Abelian p-subgroup. Let D be a derived subgroup of G. If ]D I = :xD, then by the inductive hypothesis G contains an infinite elementary Abelian p-subgroup. Thus, one can assume that IDI < :xD. Since G is residually finite, there exists a normal subgroup R < G of finite index such that RND = 1 and, consequently, R is an infinite Abelian subgroup of G. As was shown above, R contains an infinite elementary Abelian p-subgroup. Thus, G contains an infinite elementary Abelian p-subgroup. This contradicts Lemma 1. Therefore, the group G is finite. The lemma has been proved. Lemma 3. Let G be a residually finite group, and let H be an almost metanilpotent soluble group of finite rank. If ~(G) C F(H), then G is an almost metanilpotent group. Proof. Since G is a residually finite group, we see that G is a subgroup of the Cartesian product = ~iGi, where Gi E JZ(G). The group H has the normal series