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By Lucchhini A.

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J. Wong, These proceedings. A CHARACTERIZATION OF W. J. Wong 1 We consider finite groups G with the following property. (*) Lln L If q G 2 has subgroups = < t > L , L 2 l ,where t is odd, the simple group property, with ql = q2 = q . has order 2, PSP4(q) q has this In this case we write a 4-dimensional vector space field of such that V over the elements, with a non-degenerate 1. Research partially supported by National Science Foundation grant GP-6652 at the University of Notre Dame. 31 32 FINITE GROUPS alternating bilinear form, as an othogonal direct sum of two hyperbolic planes L , L 2 l as the subgroups of t~e ing to correspond- PSP4(q) subgroups of the symplectic group leaving fixed the vectors of SP4 (q) = Sp (V) V ' V 2 l and take V ' V ' 2 l respectively.

L orbital character I(a a in such a way that @ for all (aG) assigns to " n I aG E ~. (a)} 11:= 91 ~ l I by (G E @) be subgroups of the per- Define the support of ~ U and define a corresponding numerical relative support function s(91:U) 1~(91:U) I . 58 FINITE GROUPS The concept of an orbital character directly generalizes the notion of the permutation character _ ~ if we set -1 = e. lll(cx) = [et}, et EO, we get The orbital characters have several nice o:roperties: Each ~on-negative integer values, and i > 1.

Wong's theorem Theorem. Let G be a finite group satisfy- ing the following conditionr there exist volutions a 1' one mapping a 2 , ... , n from CO a U k=l theoretic union in that and A ) m of n CA G (~k) m onto and a --- isomorphic to A ~ n) Then if in- one-to- (the setn U CG (a k ) k=l induces an isomorphism between CG (a ) (1 < k k n m >8 = C A m such (a~) k G is with A m , m For the identification of G(n, r) we used this theorem, the proof of which is due to the idea of D. Held [1] and [2]. 54 FINITE GROUPS REFERENCES 1.

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A 2-generated just-infinite profinite group which is not positively generated by Lucchhini A.

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