Nondivisibility Among Irreducible Character Co-Degrees

Abstract

For a character $\chi $ of a finite group G, the number $\chi ^c(1)={[G:{\textrm {ker}}\chi ]}/{\chi (1)}$ is called the co-degree of $\chi $ . A finite group G is an ${\textrm {NDAC}} $ -group (no divisibility among co-degrees) when $\chi ^c(1) \nmid \phi ^c(1)$ for all irreducible characters $\chi $ and $\phi $ of G with $1< \chi ^c(1) < \phi ^c(1)$ . We study finite groups admitting an irreducible character whose co-degree is a given prime p and finite nonsolvable ${\textrm {NDAC}} $ -groups. Then we show that the finite simple groups $^2B_2(2^{2f+1})$ , where $f\geq 1$ , $\mbox {PSL}_3(4)$ , ${\textrm {Alt}}_7$ and $J_1$ are determined uniquely by the set of their irreducible character co-degrees.

Recognising Ree Groups Using the Codegree Set

“…{1, (q 2 + q + 1)(q 2 − 1)(q − 1), q 2 (q 2 + q + 1)(q − 1) 2 , q 3 (q 2 + q + 1), 4 < q 1 (mod 3) q 2 (q 2 − 1)(q − 1), q 3 (q 2 − 1), q 3 (q 2 − 1)(q − 1), q 3 (q − 1) 2 } PSL(3, q), {1, 1 3 (q 2 + q + 1)(q + 1)(q − 1) 2 , 1 3 q 2 (q 2 + q + 1)(q − 1) 2 , 1 3 q 3 (q 2 + q + 1), 4 < q ≡ 1 (mod 3) 1 3 q 2 (q + 1)(q − 1) 2 , 1 3 q 3 (q − 1)(q + 1), 1 3 q 3 (q + 1)(q − 1) 2 , 1 3 q 3 (q − 1) 2 , q (q − 1) 2 } PSU(3, q), {1, (q 2 − q + 1)(q + 1) 2 (q − 1), q 3 (q 2 − q + 1), q 2 (q 2 − q + 1)(q + 1) 2 , 4 < q −1 (mod 3) q 3 (q + 1) 2 (q − 1), q 3 (q + 1) 2 , q 2 (q + 1) 2 (q − 1), q 3 (q − 1)(q + 1)} PSU(3, q), {1, 1 3 (q 2 − q + 1)(q + 1) 2 (q − 1), 1 3 q 3 (q 2 − q + 1), 1 3 q 2 (q 2 − q + 1)(q + 1) 2 , 4 < q ≡ −1 (mod 3) 1 3 q 3 (q + 1) 2 (q − 1), 1 3 q 3 (q + 1) 2 , 1 3 q 2 (q + 1) 2 (q − 1), 1 3 q 3 (q − 1)(q + 1), q (q + 1) 2 } 2 G 2 (q), q = 3 2 f +1 {1, q 3 (q 2 − 1), (q 2 − 1)(q 2 − q + 1), q 2 (q 2 − 1), q 3 (q − 1),…”

“…This conjecture was shown to hold for PSL (2, q) in [4]. In [1], the conjecture was proven for 2 B 2 (2 2 f +1 ), where f ≥ 1, PSL (3,4), Alt 7 and J 1 . The conjecture also holds in the cases where H is M 11 , M 12 , M 22 , M 23 or PSL (3,3) by [9].…”

On the Characterization of Alternating Groups by Codegrees

“…Then G ∼ = H. This conjecture appears in the Kourovka Notebook of Unsolved Problems in Group Theory as question 20.79 [18]. It has been verified for PSL(2, q), PSL (3,4), Alt 7 , J 1 , 2 B 2 (2 2f +1 ) where f ≥ 1, M 11 , M 12 , M 22 , M 23 and PSL (3,3) by [1,3,13]. The conjecture has also been verified for PSL (3, q) and PSU (3, q) in [19] and 2 G 2 (q) in [14].…”

“…then the restriction of V λ to A n splits into two irreducible representations of the same dimension, so dim(U λ ) = 1 2 dim(V λ ). In this case,…”

“…Let G be a finite group and Irr(G) the set of all irreducible complex characters of G. For any χ ∈ Irr(G), define the codegree of χ as cod(χ) := |G:ker(χ)| χ (1) . Then define the codegree set of G as cod(G) := {cod(χ) | χ ∈ Irr(G)}.…”