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 -separable groups, for sets of primes pi , by proving the existence of a conjugacy class of subgroups in pi -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 -separable groups by Čunihin, regarding existence and properties of Hall subgroups.
Highlights
The well-known result of Carter [7] states that each soluble group possesses exactly one conjugacy class of self-normalizing nilpotent subgroups
This is the origin of the theory of distinguished conjugacy classes of subgroups in finite soluble groups related to certain classes of groups, which quickly splits into the theories of covering subgroups and projectors related to Schunck classes and formations, and the dual theory of injectors and Fitting classes. (See Definitions 3.1, 3.2.) We refer to the excellent monographs [5, 9] for an account of developments on the topic in the universes of soluble and finite groups
If π is a set of primes, π-separable groups have Hall π-subgroups, and every π-subgroup is contained in a conjugate of any Hall π-subgroup, by a well-known result of Cunihin [6]
Summary
The well-known result of Carter [7] states that each soluble group possesses exactly one conjugacy class of self-normalizing nilpotent subgroups (the so-called Carter subgroups) His discovery was of interest at the time by analogy with Cartan subalgebras, and by sharing the flavour of Hall’s result on the existence and conjugacy of Hall ρsubgroups, for all sets of primes ρ, in every finite soluble group. In [2] a definition of F-normality in soluble groups consistent with the lattice properties of F-subnormal subgroups is achieved This concept is applied in this paper to extend Carter subgroups taking heed of its very definition, as nilpotent self-normalizing subgroups, and enables us to address the lack of a Carter and Gaschütz counterpart in the extension of Hall theory from soluble groups to π-separable groups. In a forthcoming paper [4], our Carter-like subgroups are used to generalize these results to π-separable groups
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