A class of Frobenius regular groups

on 32-Transitive Frobenius Regular Groups

“…Previous contributions to the study of Z(/c, m)-groups include [4], [5], [8], [14] and [16]. Since the soluble (k, m)-groups and the Zassenhaus groups are known, [19], [20], [6; p. 378] the determination of all Z(k, m)-groups is completed by our main result.…”

“…To conclude this section we note that (C) may fail even for a (non-regular) primitive group. D. F. Holt has observed that if the prime p is sufficiently large then the group PSL (2, p) acting on the set of cosets of a maximal 4 4 does not satisfy (C), because the set of elements with fixed points is too small to be transitive.…”

“…In proving the theorem we may assume that k > 2 so that m is odd [24]. It follows (by considering the action of an involution on the design) that the size n = (k -l)m + 1 of fi is divisible by k and hence m = kt +1 for some positive integer t. These facts, which can be found in [4], [5] and [14], play an important role, in the exceptional character theory.…”

“…(b) If t = 1 then the underlying design is clearly affine. But G is line-transitive [4], so it contains the translation group and is not simple [25].…”