Induced characters with equal degree constituents

Abstract

We study the situation of a finite group

Induced characters of equal degree constituents

“…Since N C is a direct product of N and C, and C ≤ ker(θ), we may write φ = θ N ∈ Irr(N ) and observe that φ extends to χ with C ≤ ker(χ). The proof of Proposition 4.2 of [5] shows…”

“…and suppose that C is nonsolvable. By Theorem 4.3 of [5], there exists a nonprincipal character λ ∈ Irr(C) that has an extension to an irreducible character χ of G. Let φ = χ H and observe that φ is an extension of λ. Now φ extends to χ ∈ Irr(G) and φ C = λ ∈ Irr(C).…”

“…Dr. Stephen Gagola and Dr. Sezgin Sezer wrote a paper studying a character theoretic hypothesis (denoted ( * )) [5]. We seek to extend their work.…”

“…Since N C/C ∼ = N we may use Proposition 4.2 of [5] to find a nonprincipal irreducible character θ ∈ Irr(N C/C) that extends to an irreducible character of Aut(N ). Henceθ extends to some χ ∈ Irr(G/C).…”

“…We mimic the proof of Lemma 4.1 of [5] with slight modifications. If S is one of the 26 sporadic simple groups or if S = 2 F 4 (2) , then select φ ∈ Irr(S) # of smallest degree such that φ is invariant in Aut(S) and hence φ extends to Aut(S) as Aut(S)/S is cyclic.…”