A Conjecture of Frobenius and the Sporadic Simple Groups, II

Abstract

A conjecture of Frobenius which has been reduced to the classification of finite simple groups is verified for the sporadic simple groups. Let G be a finite group and n be a positive integer dividing IGI. Let Ln(G) = {x E G xn = 1). Then by a theorem of Frobenius [6] one knows that ILn(G)I = cnn for some integer cn. Frobenius conjectured that Ln(G) forms a subgroup of G provided ILn(G)I = n (see [2]). Zemlin [25] has reduced the conjecture to the classification of finite simple groups which is now complete (see [8]). The author has verified the conjecture for the Fischer Griess monster F1 and the Fischer baby monster F2 in [24]. The purpose of this note is to prove the following THEOREM. The conjecture of Frobenius is true for all the sporadic simple groups. The proof of our theorem has been carried out in the following way with the use of a computer. Let G be one of the sporadic simple groups. By [24] we may assume that G # F1 and G # F2. Let f(G, t) be the number of elements of order t in G and Ord(G) = {order of x I x E G}. Tables of f(G, t) are given in the Appendix; see the supplements section at the end of this issue. For f(G, t) the reader is referred to the following papers: Ml1, M22, M23 Burgoyne and Fong [1] M12, M24 Frobenius [5] ii Janko [14] HJ = J2 Hall and Wales [8] HJM= J3 Janko [15] J4 Janko [16] HiS Frame [4] Suz Wright [23] McL, .3 Finkelstein [3] Rud Rudvalis [19] HHM Held [10] LyS Lyons [17] Received June 21, 1982; revised March 19, 1984 and July 11, 1984. 1980 Mathematics Subject Classification. Primary 20D05.