Nonsolvable Finite Groups Whose All Nonsolvable Superlocals Are Hall Subgroups
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OLVABLE FINITE GROUPS WHOSE ALL NONSOLVABLE SUPERLOCALS ARE HALL SUBGROUPS V. A. Vedernikov
UDC 512.542
Abstract: We describe the nonabelian simple finite groups whose every nonsolvable local maximal subgroup is a Hall subgroup, and the nonsolvable finite groups whose all nonsolvable superlocals are Hall subgroups. DOI: 10.1134/S003744662005002X Keywords: finite group, nonsolvable group, local maximal subgroup, superlocal, Hall subgroup
1. Introduction Only finite groups are considered. Thompson described in [1] the structure of N -groups; i. e., the nonsolvable groups whose every local subgroup is solvable. Monakhov studied in [2] the structure of π-solvable groups with maximal Hall subgroups whose indices in the group are π-numbers. Tikhonenko and Tyutyanov described in [3] all nonabelian simple groups modulo the classification of finite simple groups, and Maslova described in [4] all nonabelian simple composition factors of every nonsolvable group with maximal Hall subgroups. Maslova and Revin obtained in [5] a full description of the structure of finite groups whose every maximal subgroup is a Hall subgroup. Modulo the classification of finite simple groups, the author described in [6] the structure of nonabelian simple groups G whose every maximal subgroup is either a solvable group or a Hall subgroup of G, as well as the structure of nonabelian composition factors of every nonsolvable group whose every nonsolvable subgroup is a Hall subgroup. There are some other articles in this direction; see [7–10] for instance. A subgroup H of a group G is called a local (p-local) subgroup of G whenever G includes a nonidentity primary subgroup (p-subgroup) P such that H = NG (P ). A subgroup H of a group G is called a local (p-local) maximal subgroup of G whenever H is both local (p-local) and maximal in G. A subgroup H of a group G is called a nonsolvable maximal subgroup of G whenever H is a nonsolvable subgroup of G and H is maximal in G. A subgroup H of a group G is called a nonsolvable local (p-local) maximal subgroup of G whenever H is local (p-local) in G and H is a nonsolvable maximal subgroup of G. A subgroup H of a group G is called a maximal local (p-local) subgroup of G whenever H is inclusion-maximal in the set of all local (p-local) subgroups of G. In every nonabelian simple group G each local (p-local) maximal subgroup is a maximal local (p-local) subgroup, but the converse is false. Following [11, 1.5], denote by Chev(p) the set of all groups of Lie type over finite fields of characteristic p. Each maximal p-local subgroup in G ∈ Chev(p) is a parabolic subgroup of G by [11, Theorem 1.41]. Aschbacher introduced [12] the concept of superlocal to generalize parabolic subgroups: A p-superlocal in a group G is a p-local subgroup A such that A = NG (Op (A)), and A is called a superlocal in G whenever A is a p-superlocal in G for some p ∈ π(A). As established in [12], each p-local subgroup H of G lies in some p-superlocal A of G such that Op (H) ≤ Op (A). Following [13], define the binary relation ≤p on the set of all subg
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