On local-solvability of groups containing a locally nilpotent maximal subgroup

Abstract

i, Introduction. In this paper we study groups with a locally nilpotent maximal subgroup. Our purpose is to find which conditions on the group G, or on the maximal locally nilpotent subgroup M, ensure the local solvability of G. tn the theory of finite groups there is a well-known result of Thompson, which says that if G is a finite group with a maximal subgroup M which is nilpotent and of odd order, then G is solvable (see [9]). The restriction on the order of M was due to the fact that, in order to obtain the above result, Thompson uses his famous J Theorem to prove the p-nilpotency of G/M~ (i,e. the existence of a normal p-complement) for the primes p involved in M / M G (M o is the core of M in G). Thus p must be # 2. Deskins and Janko proved that G is solvable even when the order of M is even, provided the Sylow 2-subgroup of M has class at most 2 (see [2] and [6]), The general concern with the Sylow 2-subgroup of M is due to the existence of a counterexample: namely PSL (2, I7), which is simple and contains a maximal nilpotent subgroup isomorphic to D16, hence a 2-subgroup of nilpotency class 3. In the case that G is an infinite periodic linear group contained in GL (n, K), Wehrfritz proved that if M is a maximal subgroup of G which is locally nilpotent and if the Sytow p-subgroup of M is finite or regular for p = char K, p 4= 2, while the Sylow 2-subgroup is nilpotent Of class at most 2, then G is solvable ([I0], 12,8). Wehrfritz says that the hypothesis on the Sylow p-subgroup of M, for p char K, is probably redundant: we prove this in Section 3, Some of the above results were extended to locally finite groups by Bruno and Schuur (see [1]),