Carter and Gaschütz theories beyond soluble groups

Abstract

Classical results from the theory of finite soluble groups state that Carter subgroups, i.e. self-normalizing nilpotent subgroups, coincide with nilpotent projectors and with nilpotent covering subgroups, and they form a non-empty conjugacy class of subgroups, in soluble groups. This paper presents an extension of these facts to $$\pi $$ π -separable groups, for sets of primes $$\pi $$ π , by proving the existence of a conjugacy class of subgroups in $$\pi $$ π -separable groups, which specialize to Carter subgroups within the universe of soluble groups. The approach runs parallel to the extension of Hall theory from soluble to $$\pi $$ π -separable groups by Čunihin, regarding existence and properties of Hall subgroups.

Injectors in $$\pi $$-Separable Groups

“…We prove next that P * /W is self-N π -Dnormalizing in G/W ; in particular we will have that P * /W is an N π -maximal subgroup of G/W by [1,Corollary 4.11]. Assume that P * /W is N π -Dnormal in H/W ≤ G/W .…”

“…We observe that N π is a saturated formation and appeal to the concept of N π -Dnormal subgroup. We refer to [1,2] for the concept of G-Dnormal subgroup for general saturated formations G, and specialize this definition to our particular saturated formation N π next. For notation, if ρ is a set of primes and G is a group, Hall ρ (G) denotes the set of all Hall ρ-subgroups of G. If p is a prime, then Syl p (G) stands for the set of all Sylow p-subgroups of…”

“…self-normalizing nilpotent subgroups, in soluble groups are the cornerstone in the proof of the above-mentioned theorem of Fischer, Gaschütz and Hartley. In a previous paper [1] we initiated the study of an extension of the theory of soluble groups to the universe of π-separable groups, π a set of primes. We analyzed the reach of π-separability further from soluble groups, by means of complement and Sylow bases and Hall systems, based on the remarkable property that π-separable groups have Hall π-subgroups, and every π-subgroup is contained in a conjugate of any Hall π-subgroup.…”