Representations of Solvable Groups

Manz, Olaf; Wolf, Thomas R.

doi:10.1017/cbo9780511525971

Cited by 294 publications

(329 citation statements)

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“…Hence, G=K G P has only one non-linear irreducible character, say w, PnP 0 ¼ uðwÞ and uðwÞ consists of three conjugacy classes of P. Now we regard w as an irreducible character of G. Then, since K c kerðwÞ, KðPnP 0 Þ ¼ uðwÞ consists of three conjugacy classes of G and each element in KðPnP 0 Þ is a 2-element. This implies that every element in PnP 0 acts fixed-point freely on K. Therefore, since P is a 2-group of class 2, we conclude by [13,Lemma 19.1] that P G Q 8 and G is a Frobenius group with kernel K and complement Q 8 . Observe that there exists w A Irr 1 ðGÞ such that uðwÞ consists of four conjugacy classes of G, a contradiction.…”

Section: 1])mentioning

confidence: 75%

Finite groups whose irreducible characters vanish on at most three conjugacy classes

Zhang¹,

Shi²,

Shen³

2010

Journal of Group Theory

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Section: 1])mentioning

confidence: 75%

Finite groups whose irreducible characters vanish on at most three conjugacy classes

Zhang¹,

Shi²,

Shen³

2010

Journal of Group Theory

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“…We assume that the reader is familiar with the basic properties of p-special and p-factorable characters of p-separable groups (see [4] or [10] for details).…”

Section: Definitions and Previous Resultsmentioning

confidence: 99%

“…Note that by the definition of inductive pairs and [10,Theorem 21.7], if ðV ; gÞ is an inductive pair for the p-chain N, then g is factorable.…”

Section: Definitions and Previous Resultsmentioning

confidence: 99%

Inductive pairs and lifts in solvable groups

Cossey

Lewis

2011

Journal of Group Theory

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Abstract. The Fong-Swan theorem showed that for each irreducible Brauer character j of a finite solvable group, there exists at least one ordinary irreducible character w that lifts j. Recently there has been some progress toward understanding the set of lifts of a given Brauer character, by constructing certain canonical sets of lifts and studying their properties. In this paper we develop the theory of lifts of Brauer characters that are induced from inductive pairs. This leads to a lower bound on the number of 'well-behaved' lifts of a given Brauer character in terms of the inductive pair. Moreover, it is shown that given a 'well-behaved' lift, the vertices for the lift (a generalization of the vertices of the corresponding Brauer character) are unique up to conjugacy.

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“…This graph was intro-duced in [5]. In particular, in [5] it is proved that if G is solvable then DðGÞ has at most two connected components.…”

Section: Lemma 1 ([1 Corollary 24])mentioning

confidence: 99%