A Characterization of Conway's Group .3 by Daniel Fendel

THEOREM. Let G be a finite group and s an involution in G, such that CG(S)~CGQ(SO)Assume G^Co(s)0(G) (0(G) is the maximal normal odd order subgroup of G). Then Gc^Go. In particular, G has the following properties : (i) G has order 2 • 3 • 5 • 7 • 11 • 23, and is simple. (ii) G has two conjugacy classes of involutions. One class is represented by the involution s. A representative t of the second class has centralizer C G ( / ) — ( 0 X M i 2 (MYL is the Mathieu group). (iii) The normalizer of a Sylow 23-subgroup is a Frobenius group of order 1 1 2 3 . (iv) The normalizer of a Sylow 11-subgroup is a direct product of Z2 (the group of order 2) and a Frobenius group of order S • 11. (v) The normalizer of a Sylow 7-subgroup is a direct product of Sym3 (the symmetric group) and a Frobenius group of order 6 • 7 with kernel of order 7. (vi) A Sylow 5-subgroup is nonabelian of exponent 5. There are two classes of elements of order 5, with centralizers of orders 2 • 3 • 5 and 2 • 3 • 5. The normalizer of a Sylow S-subgroup has order 2 • 3 • 5. (vii) G has no outer automorphisms. The character table of G is obtained in the course of the proof, and will appear elsewhere with details of the proof.