Groups whose homomorphic images have a transitive normality relation

Robinson, Derek J. S.

doi:10.1090/S0002-9947-1973-0323907-0

Groups whose homomorphic images have a transitive normality relation
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by Derek J. S. Robinson PDF

Trans. Amer. Math. Soc. 176 (1973), 181-213 Request permission

Abstract:

A group G is a T-group if $H \triangleleft K \triangleleft G$ implies that $H \triangleleft G$, i.e. normality is transitive. A just non-T-group (JNT-group) is a group which is not a T-group but all of whose proper homomorphic images are T-groups. In this paper all soluble JNT-groups are classified; it turns out that these fall into nine distinct classes. In addition all soluble $JN\bar T$-groups and all finite $JN\bar T$-groups are determined; here a group G is a $\bar T$-group if $H \triangleleft K \triangleleft L \leq G$ implies that $H \triangleleft L$. It is also shown that a finitely generated soluble group which is not a T-group has a finite homomorphic image which is not a T-group.

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