2021
DOI: 10.48550/arxiv.2102.07132
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Normal $p$-complements and irreducible character codegrees

Abstract: Let G be a finite group and p ∈ π(G), and let Irr(G) be the set of all irreducible complex characters of G. Let χ ∈ Irr(G), we write cod(χ) = |G : kerχ|/χ(1), and called it the codegree of the irreducible character χ. Let N G, write Irr(G|N ) = {χ ∈ Irr(G) | N kerχ}, and cod(G|N ) = {cod(χ) | χ ∈ Irr(G|N )}. In this Ipaper, we prove that if N G and every member of cod(G|N ′ ) is not divisible by some fixed prime p ∈ π(G), then N has a normal p-complement and N is solvable.

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