On homomorphic images of locally graded groups (Q1332496)

A group \(G\) is locally graded if every non-trivial finitely generated subgroup of \(G\) has a finite non-trivial image. Clearly the class of locally graded groups is not closed under forming homomorphic images. In this paper the author proves that, if \(G\) is a locally graded group and \(H\) is a normal subgroup of \(G\), then also the factor group \(G/H\) is locally graded provided that \(H\) satisfies one of the following conditions: (i) \(H\) is abelian and the normal closure in \(G\) of each element of \(H\) is finitely generated; (ii) \(H\) has an ascending series of \(G\)-invariant subgroups whose factors are abelian with finite torsion- free rank; (iii) \(H\) is contained in the FC-hypercentre of \(G\) and has no section isomorphic to the direct product of infinitely many finite non- abelian simple groups. In particular, \(G/H\) is locally graded if \(G\) is locally graded and \(H\) is a normal subgroup contained in the hypercentre of \(G\).