1. (i) Suppose K is a conjugacy class of Sn contained in An; then K is called split if K is a union of two conjugacy classes of An. Show that the number of split conjugacy classes contained in An is equal to the number of characters χ ∈ Irr(Sn) such that χAn is not irreducible. (Hint. Consider the vector space of class functions on An which are invariant under conjugation by the transposition (12).)

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Character Theory of Finite Groups

Element orders and character codegrees

Let G be a finite group and be a character of G. The codegree of is . We write for the set of prime divisors of , for the set of prime divisors of and . We show that . This improves the main result of Yang and Qian [‘The analog of Huppert’s conjecture on character codegrees’, J. Algebra 478 (2017), 215–219].

On analogues of huppert’s conjecture

The codegree of an irreducible character χ of a ﬁnite group G is | G : ker χ | /χ (1). We show that the Ree group 2 G 2 ( q ), where q = 3 2 f + 1 , is determined up to isomorphism by its set of codegrees.

RECOGNISING REE GROUPS 2 G 2 ( q ) USING THE CODEGREE SET

Abstract The codegree of an irreducible character $\chi $ of a finite group G is $|G : \ker \chi |/\chi (1)$ . We show that the Ree group ${}^2G_2(q)$ , where $q = 3^{2f+1}$ , is determined up to isomorphism by its set of codegrees. 

/pdf/recognising-ree-groups-using-the-codegree-set-1g56odt4.pdf

Recognising ree groups using the codegree set

Let G be a finite group and the codegree of irreducible character χ of group G is defined as |G:ker(χ)|/χ(1). In this paper, we prove that if a nonsolvable group G has exactly five character codegr...

Yang Liu

Papers

Nonsolvable groups with five character codegrees