A conjecture of Frobenius and the sporadic simple groups. II

Abstract

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 « be a positive integer dividing \G\. Let Ln(G) = {x e G\x" = 1). Then by a theorem of Frobenius [6] one knows that \Ln(G)\ = cnn for some integer cn. Frobenius conjectured that Ln(G) forms a subgroup of G provided \Ln(G)\ = « (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 Fx 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 + Fx and G + F2. Let f(G, t) be the number of elements of order t in G and Ord(G) = (order of x \ 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: We look for all the subsets ß of Ord(G) satisfying the following two conditions: (a) If t is a member of ß, then ß contains all divisors of t. In particular 1 is always a member of ß.(b) For the subset ß of Ord(G) in (a), E,eS2/(G, /) divides |G|.A PASCAL program effectively generates such a subset ß with the use of a recursive concept. Special add and divide routines are used for explicit calculation with very large digit numbers. Then we have only six possibilities for ß:(i) ß = {1}, (ii) ß = Ord(G), (iii) ß = (1,2,3,4,5,10,11} and G is the Mathieu group MX2, (iv) ß = (1,2,3,5,7,11,19} and G is the Janko group Jx,