Primes dividing character degrees and character orbit sizes

Gluck, David

doi:10.1090/S0002-9939-1987-0902531-8

Primes dividing character degrees and character orbit sizes
HTML articles powered by AMS MathViewer

by David Gluck PDF

Proc. Amer. Math. Soc. 101 (1987), 219-225 Request permission

Abstract:

We consider an abelian group $A$ which acts faithfully and coprimely on a solvable group $G$. We show that some $A$-orbit on $\operatorname {Irr}(G)$ must have cardinality divisible by almost half the primes in $\pi (A)$. As a corollary, we improve a recent result of I. M. Isaacs concerning the maximum number of primes dividing any one character degree of a solvable group.

References

David Gluck, Trivial set-stabilizers in finite permutation groups, Canadian J. Math. 35 (1983), no. 1, 59–67. MR 685817, DOI 10.4153/CJM-1983-005-2
David Gluck and Thomas R. Wolf, Defect groups and character heights in blocks of solvable groups. II, J. Algebra 87 (1984), no. 1, 222–246. MR 736777, DOI 10.1016/0021-8693(84)90168-6
B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703
I. Martin Isaacs, Character theory of finite groups, Pure and Applied Mathematics, No. 69, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0460423
I. M. Isaacs, Solvable groups character degrees and sets of primes, J. Algebra 104 (1986), no. 2, 209–219. MR 866771, DOI 10.1016/0021-8693(86)90211-5
Thomas R. Wolf, Defect groups and character heights in blocks of solvable groups, J. Algebra 72 (1981), no. 1, 183–209. MR 634622, DOI 10.1016/0021-8693(81)90317-3
Thomas R. Wolf, Solvable and nilpotent subgroups of $\textrm {GL}(n,\,q^{m})$, Canadian J. Math. 34 (1982), no. 5, 1097–1111. MR 675682, DOI 10.4153/CJM-1982-079-5