If H is a subgroup of the group G then we denote by H_G the normal core of H in G i.e. the intersection of all conjugates of H in G. Several papers have dealt with so called “CF-groups”, i.e. groups G in which the factor group H/H_G is finite for all subgroups H of G. For example it has been proved that if G is a locally finite CF-group, then there exists a normal abelian subgroup of G of finite index in G and there is a positive integer n such that H/H_G has order at most n; furthermore G has an abelian normal subgroup of index bounded in terms of n only. Groups G that satisfy the hypothesis that H/H_G has finite (Prufer) rank for all subgroups H have been also considered, and a corresponding result holds, locally soluble-by-finite groups with the previous property are abelian-by-finite rank, i.e. there exists a normal abelian subgroup A of G with G/A of finite rank. In this paper we study groups in which, for all subgroups H of G, H/H_G satisfies min, the minimal condition on subgroups, or max, the maximal condition on subgroups. We show that, with the additional hypothesis that G has all of its periodic images locally finite, G has an abelian normal subgroup A such that G/A has min; further consequences are then established. With the maximal condition replacing the minimal condition, a similar conclusion does not hold: we give an example of a (torsionfree) nilpotent group G such that H/H_G satisfies max for all subgroups H, but G is not abelian-by-max.
Groups with H/core(H) satisfying max or min for all subgroups H
LONGOBARDI, Patrizia;MAJ, Mercede;
2004-01-01
Abstract
If H is a subgroup of the group G then we denote by H_G the normal core of H in G i.e. the intersection of all conjugates of H in G. Several papers have dealt with so called “CF-groups”, i.e. groups G in which the factor group H/H_G is finite for all subgroups H of G. For example it has been proved that if G is a locally finite CF-group, then there exists a normal abelian subgroup of G of finite index in G and there is a positive integer n such that H/H_G has order at most n; furthermore G has an abelian normal subgroup of index bounded in terms of n only. Groups G that satisfy the hypothesis that H/H_G has finite (Prufer) rank for all subgroups H have been also considered, and a corresponding result holds, locally soluble-by-finite groups with the previous property are abelian-by-finite rank, i.e. there exists a normal abelian subgroup A of G with G/A of finite rank. In this paper we study groups in which, for all subgroups H of G, H/H_G satisfies min, the minimal condition on subgroups, or max, the maximal condition on subgroups. We show that, with the additional hypothesis that G has all of its periodic images locally finite, G has an abelian normal subgroup A such that G/A has min; further consequences are then established. With the maximal condition replacing the minimal condition, a similar conclusion does not hold: we give an example of a (torsionfree) nilpotent group G such that H/H_G satisfies max for all subgroups H, but G is not abelian-by-max.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
