By Irving Reiner

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**Additional info for Representation Theory of Finite Groups and Related Topics: Proceedings of the Symposia in Pure Mathematics-Madison, Wis.-1970**

**Example text**

From the point of view of Chevalley groups, however, the hypothesis is somewhat artificial. One would prefer a hypothesis which somehow "captures" Sylow 2-subgroups of all Chevalley groups of odd characteristic. BIBLIOGRAPHY 1. G. Glauberman. wwcteristic subgroup ora p-stable group, Canad. J. Math. 20 (1968),11011135. MR37 #6365. 2. J. G. Thompson, Faclorizalions ofp-solmble groups, Pacific J. Math. 16 (1966),371-372. MR 32 #5735. 3. J. finite groups with abelian SylO\I' 2-subgroups, Ann. of Math.

Then IGI = LgEGZ(g - 1 )z(g). UNIVERSITY OF UTAH The Number of Conjugacy Classes in a Finite Group P. X. Gallagher Let k(G) denote the number of conjugacy classes (= number of irreducible characters) of a finite group G. Ernest [IJ has shown that for each subgroup H, (1) k(G) ~ (G:H)k(H). If there is equality in (I), then H is a normal subgroup. Sah [3J has shown that for each normal subgroup N. the equation (2) k(G) = IG/NI· k(N) is equivalent to each of the foHewing: (a) G = C(a)N, for each a E G; (b) G/N is abelian, and each character of N extends to a character of G.

Sci. Eco1e Norm. Sup. (4) 1 (1968), '11-148. MR 38 #217. 5. J. Riordan. An introduction to combinatorial ana/vsis. Wiley. New York; Chapman & Hall, London, 1958, pp. 34-38. MR 20 # 3077. 6. L. Solomon, The Burnside algebra olafinite group, J. Combinatorial Theory 2 (1967),603-615. MR 35 # 5528. UNIVERSITAT BIELEFELD, GERMANY Some Decomposable Sylow 2-Subgroups and a Nonsimplicity Condition Paul Fang Let G be a finite group with Sylow 2-subgroup P. Assume the elements of Pare fused as in a direct product; what are sufficient conditions for the nonsimplicity of G?