The existence and uniqueness of injectors for Fitting sets of solvable groups

Abstract

Abstract. A short alternative proof is given of the existence and uniqueness of injectors in a Fitting set which avoids use of the Carter subgroup, a concept from the dual theory of projectors.A Fitting set X of a group G is a non-empty set of subgroups which is closed under normal product, subnormal inheritance, and conjugation in G. If H ≤ G, then the groups from X which are subgroups of H form a Fitting set of H which, since there is seldom danger in confusion, we also call X . An X-injector S of G is a subgroup with the property that S ∩ H is X -maximal in H for all H ¢ G. If G is solvable, then X -injectors exist uniquely up to conjugacy.This theorem was first proved by Fischer, Gaschütz and Hartley in [3]. Their proof can also be found in [2], which also provides a comprehensive introduction to solvable group theory. The core of the proof is a clever lemma of Hartley which uses Carter subgroups (projectors for the class of nilpotent groups).It is unsatisfying to have the proof of this fundamental result dependent on the corresponding result from the dual theory of projectors. We would prefer a proof based on more elementary ideas. Unfortunately alternative proofs [1] are often long, and our preference is not strong grounds for discarding a short proof in favor of a longer one. Here, however, is a short and simple proof of this type.Recall that a p-subgroup P 0 ≤ P ∈ Syl p (G) is strongly closed in P if P g 0 ∩P ≤ P 0 for all g ∈ G. The proof uses two simple lemmas. The first lemma states that strongly closed p-subgroups of solvable groups are normally embedded. This is well known (for example, see [4]). The second lemma is a special case of Corollary I(7.11) in [2], originally due to Schaller. Neither lemma is particularly difficult to prove. Lemma 1. Let G be a solvable group. Let P 0 be strongly closed in P ∈ Syl p (G). Then there exists N ¢ G with N ∩ P = P 0 .Proof. Induct on |G|. Let 1 = M ¢ G and let H denote HM/M for H ≤ G. Since P 0 is strongly closed in P with respect to G, by hypothesis P 0 = P ∩ N for N ¢ G. Hence there exists N ¢ G with N ∩ P = P 0 (M ∩ P ). If O p (G) = 1 we can set M = O p (G), and we are done. If O p (G) ∩ P 0 = 1 we can set M = O p (G) ∩ P 0 , and we are done. But now P 0 ≤ C G

Инъекторы Во Множестве Фиттинга Конечной Группы

“…Докажем свойства строгой -замкнутости подгрупп, аналогичные свойствам строгой замкнутости из работы [7], которые будем использовать в дальнейшем.…”

“…Следствие 3.5 (см. [7]). Пусть --разрешимая группа и 0 строго замкнута в ∈ Syl ( ) относительно для некоторого простого числа из .…”

“…Следствие 3.7 (см. [7]). Пусть --разрешимая группа, , -простые числа из и группа 0 строго замкнута в ∈ Syl ( ) относительно , а группа 0 строго замкнута в ∈ Syl ( ) относительно .…”