Abstract
Let G be a finite group. Goldschmidt, Flores and Foote investigated the following concept: A subgroup A of S is said to be strongly closed in S with respect to G if for every a
A, every element of S that is fused in G to a lies in aG
. In particular, when A is of prime-power order and S is a Sylow subgroup containing it, A is simply said to be a strongly closed subgroup. Bianchi, Mauri, Herzog and Verardi called a subgroup H of G an -subgroup if NG
(H) Hg
H for all g
G. In fact, an -subgroup of prime power order is the same as a strongly closed subgroup. In this paper, groups with certain -subgroups of prime order are studied. For example, we give new characterizations of finite minimal non-
-groups (a -group is a group for which normality is a transitive relation) by -subgroups.
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