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By Irving Reiner

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Dade gave an example in which equality holds in (3) but not every character of N extends to a character of G. REFERENCES 1. J. A. Ernest, Central intertwining numbers for representations o/finite groups, Trans. Amer. Math. Soc. 99 (1961),499-508. MR 23 #A2467. 2. ~-, Embedding numbers/or finite groups, Proc. Amer. Math. Soc. 13 (1962),567-570. MR 25 # 1218. 3. -H. Sah, Automorphisms offinite groups. J. Algebra 10 (1968), 47-68. MR 37 # 5287. 4. I. Schur, Ober die Darstelltmgen endlichen Gruppen durch gebrochene lineare Substitutionen, J.

Furthermore Clifford's theory gives a natural one-to-onc correspondence between all irreducible \j-characters of the twisted group algebra \j[H corresponding to the extension H

Higman and G. Glauberman. Let 51, S2 be any conjugates ofjl,jz respectively, and let p be the 2-part of SIS2' Since is dihedral, there is a conjugate s~ of S2 in such that SIS~ = p. ;; P. (I) implies that (sls~)g andjlj2 are conjugate, or equivalently, p andjJ12 are conjugate. Let K;, K~ be the conjugate classes of G containingj l'iz respectively. and let K 10 ••• , Kt be the classes of G in the section ofjJ2' We have just shown that t [KlJ [KiJ = L aJKJ. j= 1 where the aj are nonnegative integers, and AMS 1969 subject classifications.

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