Fittingfunktoren in endlichen aufl�sbaren Gruppen. I

The Frattini argument and 𝑡-groups

“…The notion that a subgroup satisfies the Frattini argument was incisive to the development in §7 of [2]. Obviously, normal subgroups and maximal subgroups satisfy the strong Frattini argument.…”

“…From Beidleman, Brewster, and Hauck [2], it is easily deduced that h, k, and / are Fitting functors. We can identify I(k, a).…”

“…Let G be a solvable group. Then G is a tgroup if and only if there is L < G such that (1) L is a Hall subgroup of G, (2) G/L is a Dedekind group, and (3) each subgroup of L is normal in G.…”

“…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.…”

“…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 N\U G f{G)}.…”