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Induction and Restriction of π-Partial Characters and their Lifts

Published online by Cambridge University Press:  20 November 2018

I. M. Isaacs
I. M. Isaacs*
Affiliation:
Mathematics Department, University of Wisconsin, Madison, WI, 53706, USA e-mail: isaacs@math.wisc.edu
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Abstract

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Let G be a finite π-separable group, where π is a set of primes. The π-partial characters of G are the restrictions of the ordinary characters to the set of π-elements of G. Such an object is said to be irreducible if it is not the sum of two nonzero partial characters and the set of irreducible π- partial characters of G is denoted Iπ(G). (If p is a prime and π = p′, then Iπ(G) is exactly the set of irreducible Brauer characters at p.)

From their definition, it is obvious that each partial character φ ∊ Iπ(G) can be “lifted” to an ordinary character χ ∊ Irr(G). (This means that φ is the restriction of χ to the π-elements of G.) In fact, there is a known set of canonical lifts Bπ(G) ⊆ Irr(G) for the irreducible π-partial characters. In this paper, it is proved that if 2 ∉ π, then there is an alternative set of canonical lifts (denoted Dπ(G)) that behaves better with respect to character induction.

An application of this theory to M-groups is presented. If G is an M-group and S ⊆ G is a subnormal subgroup, consider a primitive character θ ⊆ Irr(S). It was known previously that if |G : S| is odd, then θ must be linear. It is proved here without restriction on the index of S that θ(1) is a power of 2.

Keywords

20C15
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 6 , 01 December 1996 , pp. 1210 - 1223
Copyright
Copyright © Canadian Mathematical Society 1996

References

1. Isaacs, I. M., Primitive characters, normal subgroups and M-groups, Math. Zeit. 177(1981), 267–284.Google Scholar
2. Isaacs, I. M., Characters of π-separable groups, J. Algebra 86(1984), 98–128.Google Scholar
3. Isaacs, I. M., Characters of subnormal subgroups of M-groups, Arch. Math. 42(1984), 509–515.Google Scholar
4. Isaacs, I. M., Induction and restriction of π-special characters, Canad. J. Math. 38(1986), 576–604.Google Scholar
5. Isaacs, I. M., Partial characters of π-separable groups, Prog. Math. 95(1991), 273–287.Google Scholar
6. Isaacs, I. M., Characters and Hall subgroups of groups of odd order, J. Algebra 157(1993), 548–561.Google Scholar
7. Isaacs, I. M., The π-character theory of solvable groups, J. Austral. Math. Soc. (Ser. A) 57(1994), 81–102.Google Scholar
8. Isaacs, I. M., Characters and sets of primes for solvable groups, Proc. of NATO ASI: Finite and locally finite groups (1995), 347-376.Google Scholar
9. Lewis, M., Primitive characters of subgroups of M-groups, to appear.Google Scholar