Conjugacy Classes Of Involutions Research Articles

On Thompson's simple group

Recently, Thompson [9] constructed a new simple group E of order 2 15 ∘ 3 105 37 2 ∘ 13 ∘ 19 ∘ 31 = 90, 745, 943, 887, 872, 000. In particular, E contains only one conjugacy class of involutions, and if e is an involution in E then C E ( e) is a (nonsplit) extension of an extra-special 2-group of order 2 9 by A 9, the alternating group of degree 9. The aim of this paper is to prove the following result.

Index two simple groups

A group, G, is an index two simple group if G is a finite simple group whose order is of the form pm where p is a prime, ( p, m) = 1 and the index of a Sylow p-subgroup in its normalizer is 2. In this paper the theory of index two simple groups is developed and applied. In particular index two simple groups are shown to have exactly one conjugacy class of involutions and a sharp bound for the order of such a group is found as a function of the degree, n, of a nonidentity ordinary irreducible character in the principal p-block. It is shown that the only index two simple groups with n ≤ 25 are PSL(2, 5), PSL(2, 7), PSL(2, 8), PSL(2, 9), PSL(2, 11), PSL(2, 13), PSL(2, 16), PSL(2, 23) and PSL(2, 25).

The nonexistence of a certain type of finite simple group

The proof of the th eorem is arranged as follows. First some facts concerning the group C,(Z) are obtained. Then supposing the assertion of the theorem to be false it is shown that [G: K] is even, and so G most possess at least two conjugacy classes of involutions. Finally if x denotes the permutation character of the representation of G on the left cosets of N = C,(Z), xi = xJu may be evaluated on certain classes of H. By considering the action of G on its conjugates of Z, a contradiction to a certain value of ~4 is obtained,