ON HEIGHT ZERO CHARACTERS OF p-SOLVABLE GROUPS

概要

by Irr0 (B) the set of ordinary irreducible characters in B of height zero. If the defect of B is
positive, then a result of Cliff, Plesken and Weiss [1] asserts that |Irr0 (B)| ≥ 2. (See also [7].)
Now let N be a normal subgroup of G and suppose μ ∈ Irr(N). Let Irr(G|μ) be the set of
irreducible characters of G that lie over μ, and write Irr0 (B|μ) = Irr0 (B) ∩ Irr(G|μ). The aim
of this paper is to prove a relative version of the above result in case G is p-solvable.
Theorem. Let N be a normal subgroup of a p-solvable group G, and let B be a p-block
of G with defect group D such that |D| > |D ∩ N|. Let μ ∈ Irr(N) and suppose Irr0 (B|μ)  ∅.
Then |Irr0 (B|μ)| ≥ 2.
2. Proof of Theorem
Fix a prime p and let B be a p-block of a group G. Let N be a normal subgroup of G and
2. Proof of Theorem
let μ ∈ Irr(b), where b is a p-block of N. Suppose μ is an irreducible constituent of χN , where
χ ∈ Irr(B). By [8, Lemma 2.2], we have ht(χ) ≥ ht(μ). If ν is any other constituent of χN ,
then ν is G-conjugate to μ and belongs to a G-conjugate of b. Since G-conjugate blocks of N
have equal defects, the difference ht(χ)−ht(μ) is independent of the choice of the constituent
μ.
If ht(χ) = ht(μ), then the character χ is said to be of relative height zero with respect to N.
We denote by Irrμ0 (B) the set of all those characters in Irr(B) ∩ Irr(G|μ) having relative height
zero with respect to N. It is clear that χ ∈ Irr0 (B|μ) if and only if ht(μ) = 0 and χ ∈ Irrμ0 (B).
Now our main theorem is a consequence of the following more general result.
Theorem 2.1. Let N  G, where G is p-solvable and let B be a p-block of G with defect
group D such that |D| > |D ∩ N|. Let μ ∈ Irr(N) and assume Irrμ0 (B)  ∅. Then |Irrμ0 (B)| ≥ 2.
In order to prove Theorem 2.1, we need a series of preliminary results. ...

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