On the Multiplicities of the Character Codegrees of Finite Groups

Finite groups in which distinct nonlinear irreducible characters have distinct codegrees

“…Note that if 𝜃(1) ≤ 𝑝 𝑚 − 1, then Claim 1 yields 𝑍 ≥ 𝑉 and thus 𝑍 ≥ 𝑉 × 𝐸 = 𝑈, a contradiction. Consequently, 𝑝 𝑚 − 1 < 𝜃 (1). Since 𝜃(1) also divides 2(𝑝 𝑚 − 1), we have 𝜃(1) = 2(𝑝 𝑚 − 1) for all nonlinear 𝜃 ∈ Irr(𝐺∕𝑉).…”

“…Ebrahimi studies a special case of $$D^{\prime }$$‐groups in [8], where it is shown that if the irreducible complex characters (including linear characters) of a nontrivial finite group G have distinct codegrees, then G is isomorphic to either the cyclic group of order 2 or the symmetric group of degree 3. Akhlaghi, Ebrahimi, and Khatami [1] classify finite groups such that every element but one in the set of irreducible complex character codegrees is afforded by a unique irreducible character.…”

“…For an irreducible complex character 𝜒 of a finite group 𝐺, we define its codegree [15] by cod(𝜒) = |𝐺 ∶ ker 𝜒| 𝜒 (1) .…”