Geodesic
Induction and Restriction of π-Partial Characters and their Lifts
Canadian journal of mathematics, Tome 48 (1996) no. 6, pp. 1210-1223

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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.
DOI : 10.4153/CJM-1996-064-9
Mots-clés : 20C15 
Isaacs, I. M. Induction and Restriction of π-Partial Characters and their Lifts. Canadian journal of mathematics, Tome 48 (1996) no. 6, pp. 1210-1223. doi: 10.4153/CJM-1996-064-9
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