Transitive subgroups of transitivep-groups

Abstract

SHEeVERD and I proved in [4] that every transitive group of prime-power degree p" has an n-generator subgroup transitive on the same symbols. As one sees immediately from a consideration of elementary abelian groups in the right regular representation, this result is best possible; however, we shall see that in most other cases there is an (n 1)-generator transitive subgroup. At the outset we can restrict attention to p-groups, since every Sylow p-subgroup of a transitive group of degree pn is transitive [5, page 6]. To smooth things later on, let us deal with the case n = 2 first. The transitive 2-groups of degree 4 are the dihedral group of order 8 and the two groups of order 4; of these, only the four-group fails to have a cyclic transitive subgroup. For odd p, the non-abelian group of exponent p and order p 3 has a transitive representation of degree p2 but, of course, no cyclic transitive subgroup, Therefore the restriction that n be at least 3 in the following theorem cannot be removed when p is odd. The notation employed is that of [5].