### c-SECTIONS, SOLVABILITY AND LARGE SUBGROUPS OF FINITE GROUPS

BARBARA BAUMEISTER, GIL KAPLAN
2012 Bulletin of the Australian Mathematical Society
c-Sections of maximal subgroups in a finite group and their relation to solvability have been extensively researched in recent years. A fundamental result due to Wang ['C-normality of groups and its properties', J. Algebra 180 (1998), 954-965] is that a finite group is solvable if and only if the c-sections of all its maximal subgroups are trivial. In this paper we prove that if for each maximal subgroup of a finite group G, the corresponding c-section order is smaller than the index of the
more » ... mal subgroup, then each composition factor of G is either cyclic or isomorphic to the O'Nan sporadic group (the converse does not hold). Furthermore, by a certain 'refining' of the latter theorem we obtain an equivalent condition for solvability. Finally, we provide an existence result for large subgroups in the sense of Lev ['On large subgroups of finite groups' J. Algebra 152 (1992), 434-438]. 2010 Mathematics subject classification: primary 20E34; secondary 20E28, 20D05, 20D10. In this paper we study further the notion of c-sections and its connection to solvability. In particular, for a maximal subgroup M we consider the relation between the order of the c-section |Sec(M)| and the index |G : M|. By the above, if G is solvable then obviously |Sec(M)| < |G : M| for each maximal subgroup M of G. It turns out that the converse is not true. E 1.1. Let T = O'Nan, the O'Nan simple sporadic group, and let G = Aut(T ) = T : 2. We show that |Sec(M)| < |G : M| for all maximal subgroups M of G. If M = T then |Sec(M)| = 1 < |G : M| = 2. Let M be maximal in G, M T . Since T/1 is a chief factor of G and M T , M > 1, we have S := Sec(M) = M ∩ T . By G = MT it follows that for each g ∈ G there exists t ∈ T such that S g = S t . Thus Con T (S ) = Con G (S ), and so Con T (N T (S )) = Con G (N T (S )). Assume now that |Sec(M)| ≥ |G : M|. Then |S | ≥ |G : M|, implying that |S | ≥ |T : S | and |S | ≥ |T | 1/2 , that is, S is a large subgroup of T . By checking the list of maximal subgroups of T = O'Nan in [2], we deduce that S is contained in a maximal subgroup of T isomorphic to L 3 (7) : 2. Considering the maximal subgroups of L 3 (7) : 2, it follows that the only possibilities are S L 3 (7) : 2 and S L 3 (7), and in any case N T (S ) L 3 (7) : 2. By the information in [2] we deduce that Con T (N T (S )) Con G (N T (S )), contradicting our previous observation. Thus |Sec(M)| < |G : M| for all maximal subgroups M of G.