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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.

##### MSC:
 20E25 Local properties of groups 20E07 Subgroup theorems; subgroup growth 20E36 Automorphisms of infinite groups 20F16 Solvable groups, supersolvable groups
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##### References:
 [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
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