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- W2021106148 abstract "If enough subgroups of a group satisfy the Frattini argument in the group, then normality is a transitive relation within the group. Subgroup functors are used to specify what enough is. 1. Introduction. This article is an outgrowth of the investigation into functors which satisfy the Frattini argument. By a functor, we mean an association to each group G, a collection f{G) of subgroups of G such that if a: G —> G is a monomorphism, then f{Ga) = {UaU G f{G)}. Several types of functors have been explicitly formulated in the literature, e.g. Gaschutz functors in (1) and Sylow functions in (10). Of course, the idea has been implicit when associating to a group, its Sylow subgroups, its system normalizers or its /-injectors, where J is a Fitting class. In attempting to better understand the nature of injectors for Fitting classes, the notion of a Fitting functor evolved in (2). A Fitting functor is a functor / which satisfies the additional property: if N < G, then f(N) = {U n NU G f{G)}. If / is a Fitting functor and G is a group, then / satisfies the Frattini argument in G provided for each U G f(G) and each K < G, G = K ■ NG{U D K). Theorem 7.2 of (2) and Theorem 3.10 of (3) give interesting characterizations of Fitting functors which satisfy the Frattini argument in each finite solvable group. Noting the general nature of the proofs of these results, it seemed of interest to investigate the groups in which a given functor satisfies the Frattini argument. This is the context in which the groups in which normality is a transitive relation appeared. In §2 we forge this connection and in §3 we investigate functors for which our work in §2 is applicable. All groups considered are finite. Any unusual notation will be explained as it is introduced. We use F{G) for the Fitting subgroup of the group G, and Fi{G) is the subgroup" @default.
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- W2021106148 date "1987-02-01" @default.
- W2021106148 modified "2023-09-24" @default.
- W2021106148 title "The Frattini argument and $t$-groups" @default.
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