On Squares of Irreducible Characters
نویسندگان
چکیده
We study the finite groups G with a faithful irreducible character whose square is a linear combination of algebraically conjugate irreducible characters of G. In conclusion, we offer another proof of one theorem of Isaacs-Zisser. There are a few papers treating the finite groups possessing an irreducible character whose powers are linear combinations of appropriate irreducible characters, for example, [BC] and [IZ]. Our note is inspired by these two papers, especially, the second one. In what follows, G is a finite group. We use standard notation of finite group theory (see [BZ]). Recall that if χ ∈ Irr(G), then the generalized character χ (see [BZ, Chapter 4]) is defined as follows: χ(g) = χ(g) (g ∈ G). Next, Char(G) denotes the set of characters of a group G and, if θ is a generalized character of G, then Irr(θ) = {χ ∈ Irr(G) | 〈θ, χ〉 6 = 0}. The quasikernel Z(χ) of χ ∈ Char(G) is defined as follows: Z(χ) = {g ∈ G | |χ(g)| = χ(1)}. It is known that Z(χ) is a normal subgroup of G containing ker(χ) and Z(G/ ker(χ)) ≤ Z(χ)/ ker(χ) with equality if, in addition, χ ∈ Irr(G). In what follows, we use freely results stated in this paragraph. 2010 Mathematics Subject Classification. 20C15.
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