Mario Mainardis

A characterization of HN

Many of the sporadic simple groups appear to have (at least) two characteristics in the sense, for example, that there exist two prime divisors p and q of jGj such that mpðGÞd 2cmqðGÞ and, for r A fp; qg, every r-local overgroup M of a Sylow rsubgroup of G satisfies F ðMÞ 1⁄4 OrðMÞ. In contrast, hardly any other finite simple groups satisfy hypotheses of this type. Beginning with an unpublished manuscript of Gorenstein and Lyons, there has been an ongoing project to characterize the large sporadic groups (and some other simple groups) starting from hypotheses of this type. The Gorenstein–Lyons hypotheses include the assumption that q 1⁄4 2 and that m2;pðGÞd 3. This rapidly forces p 1⁄4 3. There are, however, several sporadic groups which should be regarded as groups of bicharacteristic fp; qg with fp; qg0 f2; 3g. A characterization of the Lyons group as a group of bicharacteristic f3; 5g appears in [4]. In this paper, we provide such a characterization of the Harada–Norton sporadic simple group as a group of bicharacteristic f2; 5g. Our notation and terminology is consistent with that introduced in [7]. In particular, by LpðGÞ, we mean the set of all quasisimple groups K such that K is a component of CGðxÞ=Op 0 ðCGðxÞÞ for some x A G of order p. Moreover, by ChevðpÞ we mean the set of all quasisimple groups K such that K=ZðKÞ is simple of Lie type in characteristic p. Finally, we call a finite group G K-local if every simple section of every local subgroup of G is a known finite simple group.

Radicals of

Let G be a finite group, V a complex permutation module for G over a finite G-set X , and f : V ×V → C a G-invariant positive semidefinite hermitian form on V . In this paper we show how to compute the radical V ⊥ of f , by extending to nontransitive actions the classical combinatorial methods from the theory of association schemes. We apply this machinery to obtain a result for standard Majorana representations of the symmetric groups.

S_n

We prove that a transitive permutation group of degree n with a cyclic point stabilizer and whose order is n(n-1) is isomorphic to the affine group of degree 1 over a field with n elements. More generally we show that if a finite group G has an abelian and core-free Hall subgroup Q, then either Q has a small order (2|Q|2 < |G|) or G is a direct product of 2-transitive Frobenius groups.