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A Characterization of the Tits' Simple Group

Published online by Cambridge University Press:  20 November 2018

David Parrott
David Parrott*
Affiliation:
Ohio State University, Columbus, Ohio
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In [6], J. Tits has shown that the Ree group 2F4(2) is not simple but possesses a simple subgroup of index 2. In this paper we prove the following theorem:

THEOREM. Let G be a finite group of even order and let z be an involution contained in G. Suppose H = CG(z) has the following properties:

(i) J = O2(H) has order 29and is of class at least 3.

(ii) H/J is isomorphic to the Frobenius group of order 20.

(iii) If P is a Sylow 5-subgroup of H, then Cj(P) ⊆ Z(J).

Then G = H • O(G) or G ≊ , the simple group of Tits, as defined in [6].

For the remainder of the paper, G will denote a finite group which satisfies the hypotheses of the theorem as well as G ≠ H • O(G). Thus Glauberman's theorem [1] can be applied to G and we have that 〈z〉 is not weakly closed in H (with respect to G). The other notation is standard (see [2], for example).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 4 , August 1972 , pp. 672 - 685
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Glauberman, G., Central elements in core-free groups, J. Algebra 4 (1966), 403–420.Google Scholar
2. Gorenstein, D., Finite groups (Harper and Row, New York, 1968).Google Scholar
3. Held, D., The simple groups related to M24, J. Algebra 13 (1969), 253–296.Google Scholar
4. Magnus, W., Karrass, A., and Solitar, D., Combinatorial group theory (Wiley-Interscience, New York, 1966).Google Scholar
5. Thompson, J. G., Nonsolvable finite groups all oj whose local subgroups are solvable, Bull. Amer. Math. Soc. 74 (1968), 383–437.Google Scholar
6. Tits, J., Algebraic and abstract simple groups, Ann. of Math. 80 (1964), 313–329.Google Scholar