A note on locally graded groups.

*(English)*Zbl 0852.20020A 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.

Reviewer: S.Franciosi (Napoli)

##### MSC:

20E25 | Local properties of groups |

20E07 | Subgroup theorems; subgroup growth |

20E36 | Automorphisms of infinite groups |

20F16 | Solvable groups, supersolvable groups |

##### Keywords:

finitely generated subgroups; finite images; locally graded groups; homomorphic images; Hirsch-Plotkin radical; soluble normal subgroups
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\textit{P. Longobardi} et al., Rend. Semin. Mat. Univ. Padova 94, 275--277 (1995; Zbl 0852.20020)

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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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