On normal complements of $\mathfrak{F}$-covering subgroups

Abstract

If fr is a suitably restricted formation, we show that an JF-covering subgroup H which is a Hall subgroup of the finite, solvable group G is complemented by the fF-residual of G, provided H normalizes an JF-normalizer of G. In particular, H is complemented by the SF-residual, if H is an ff-normalizer of G. Further, if $F is the class of nilpotent groups, then H complements the nilpotent residual, if G has pronormal system normalizers. Examples are given to show the necessity of the various hypotheses. In this note all groups considered are finite and solvable. The notation and definitions are essentially those of [2]. Throughout we let J be a formation which is locally induced by a class of nonempty, integrated formations 5(p), one for each prime p. If 5(p) = {1} for each prime p, then ff is the formation of nilpotent groups. Theorem A. Let $ be a formation, and let G be a finite, solvable group with ^-covering subgroup H. If (1) H is a Hall subgroup of G, and (2) H normalizes an ^-normalizer of G, then H is complemented by the ^-residual N of G. Proof. Because H covers G/N, it suffices to show that HH\N = 1. Let G be a minimal counter-example, and let A be a minimal normal subgroup of G. Suppose that H is a 7r-Hall subgroup of G, where 7r is a set of primes. Since G/A satisfies the hypotheses, it follows that HC\NgA and that NA/A is a 7r'-Hall subgroup of G/A. Therefore, A is the unique minimal normal subgroup of G. HA is a ir'-group then N would be a 7r'-Hall subgroup of G, and Hf~}N=l, a contradiction. Thus, A =H(~\N and A is a p-group for p£ir. Let 5be a p-complement of N. Using the Frattini argument, we may write G = AM, where M = N(S). If AgM, then 5 would be normal in G. Therefore A(~\M = 1, A =C(A), and M is a maximal subgroup of G. Received by the editors October 27, 1969. AMS Subject Classifications. Primary 2025, 2040, 2043, 2054.