A note on locally graded groups
A group \(G\) is locally graded if every non-trivial finitely generated subgroup of \(G\) has a non-trivial finite image. The class of locally graded groups is clearly not closed under forming homomorphic images. Thus it is interesting to know when a homomorphic image of a locally graded group is likewise locally graded. The authors prove that if \(G\) is locally graded and \(H\) is a \(G\)-invariant subgroup of the Hirsch-Plotkin radical of \(G\), then also \(G/H\) is locally graded. As a corollary it turns out, that if \(H\) is a soluble normal subgroup of the locally graded group \(G\), then \(G/H\) is locally graded.
scientific article; zbMATH DE number 889843