On Finite Groups with Cyclic Sylow Subgroups for all Odd Primes

Abstract

The purpose of this paper is to determine the structure of some finite groups in which all Sylow subgroups of odd order are cyclic. This assumption on Sylow subgroups simplifies the structure of groups considerably, but the structure of 2-Sylow subgroups might be too complicated to make any definite statement on the structure of the groups. In this paper, therefore, we shall make another assumption on 2-Sylow subgroups, and our main result may be stated as follows. Let G be a non-solvable group of finite order. We assume that all Sylow subgroups of odd order are cyclic, and moreover that a 2-Sylow subgroup is either (a) a dihedral group, or (b) a generalized quaternion group. Then G contains a normal subgroup G1 such that [G: G1] ? 2 and G1 = Z X L, where Z is a solvable group whose Sylow subgroups are all cyclic, and L is isomor-phic with the linear fractional group LF (2, p) over the prime field of characteristic p in the case (a), and with the special linear group SL (2, p) in the case (b).