PROCEEDINGSof the AMERICAN MATHEMATICAL SOCIETY Volume 106. Number 3, July 1989 CLASSIFICATION OF FINITE GROUPS WITH ALL ELEMENTS OF PRIME ORDER MARIAN DEACONESCU (Communicated by Warren J. Wong) Abstract. A finite group having all (nontrivial) elements of prime order must be a p-group of exponent p , or a nonnilpotent group of order paq , or it is isomorphic to the simple group A<. 1. Introduction Let & be the class of the finite groups having all (nontrivial) elements of prime order. Of course, & contains the /z-groups of exponent p ; but it also contains solvable groups as AA and even the simple group A5. Therefore, a description of the ^-groups is not obvious. The aim of this note is to classify these groups. Our notation is standard and conforms to that of[l]. However, we shall denote by W(G) the largest solvable normal subgroup of G. All groups are finite. We shall prove the following Main Theorem. Let G be a £P-group. Then one of the following cases occurs: I . G is a p-group of exponent p. II . (a) \G\=paq, 3<p<q, a>3, \F(G)\=p"-{, \G:G'\=p. (b) \G\=paq, 3<q<p, a>\, \F(G)\ = \G'\ = pa . (c) \G\ = 2ap, p>3, a>2, \F(G)\ = \G'\ = 2a. (d) \G\ = 2pa, p > 3, a > I, \F(G)\ = \G'\ = pa and F(G) is elementary abelian. Ill . G = A,. 2. Preliminary results For the sake of convenience, we list here some of the results used in the proof of the main theorem. 2.1 Let G be a ^-group. Then (i) Every subgroup and every factor group of G is also a .í0-group, (ii) If x G G and \x\ = p, then CG(x) is a p-group of exponent p . Received by the editors March 4, 1988, and, in revised form, November 29, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 20E34; Secondary 20D10. Key words and phrases, p-groups, soluble groups. ©1989 American Mathematical Society 0002-9939/89 $1.00+ $.25 per page 625 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use