zbMATH — the first resource for mathematics

A note on locally graded groups. (English) Zbl 0852.20020
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.

20E25 Local properties of groups
20E07 Subgroup theorems; subgroup growth
20E36 Automorphisms of infinite groups
20F16 Solvable groups, supersolvable groups
Full Text: Numdam EuDML
[1] H. Smith , On homomorphic images of locally graded groups , Rend. Sem. Mat. Univ. Padova , 91 ( 1994 ), pp. 53 - 60 . Numdam | MR 1289630 | Zbl 0817.20035 · Zbl 0817.20035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.