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For a character χ of a finite group G the number cod(χ) = |G: Ker(χ)|/χ(1) is called the codegree of χ. In this note, we verify the equivalent version of Huppert's conjecture for K3 -groups. [ABSTRACT FROM AUTHOR]

AbstractFor a character χ of a finite group G, the number cod(χ)=[G:Kerχ]χ(1) is called the codegree of χ. In this paper, we show that if for every non-principal irreducible characters χ and ϕ of G with cod(χ)≠cod(ϕ), the greatest common divisor of cod(χ) and cod(ϕ) is a prime-power, then either G is solvable or G/Sol(G) is isomorphic to one of the groups Alt5 or PSL2(8), where Sol(G) is the solvable radical of the group G. [ABSTRACT FROM AUTHOR]

In this paper we prove that some Janko groups are uniquely determined by their orders and one irreducible character degree. Also we prove that some finite simple K4-groups are uniquely determined by their character degree graphs and their orders. [ABSTRACT FROM AUTHOR]

For a character χ of a finite group G, the number cod (χ) = | G : ker (χ) | / χ (1) is called the codegree of χ. In this paper we show that if G is nonsolvable, then the codegree graph Γ (G) has a triangle. Also by using this result we show that if G has one composite codegree, then G is solvable and | cod (G) | ≤ 5. [ABSTRACT FROM AUTHOR]

For a character χ of a finite group G, the number is called the codegree of χ. We also define. In this article, we give a characterization of finite groups with at most three codegrees of irreducible characters. [ABSTRACT FROM AUTHOR]

Let G be a finite group and let cd(G) be the set of all irreducible character degrees of G. We consider finite groups G with the property that cd(G) has at most three composite members. We derive a bound 8 for the size of character degree sets of such groups. [ABSTRACT FROM AUTHOR]

Let be a finite group and let be the set of all irreducible character degrees of . We consider finite groups with the property that has at most two composite members. We derive a bound 6 for the size of character degree sets of such groups. There are examples in both solvable and nonsolvable groups where this bound is met. In the case of nonsolvable groups, we are able to determine the structure of such groups with . [ABSTRACT FROM AUTHOR]

Abstract Given a finite group G , we say that G has property P n if for every prime integer p , G has at most n − 1 irreducible characters whose degrees are multiples of p. In this paper, we classify all finite groups that have property P 4. We show that the groups satisfying property P 4 are exactly the finite groups with at most three nonlinear irreducible characters, one solvable group of order 168, SL 2 (3) , A 5 , S 5 , PSL 2 (7) and A 6. [ABSTRACT FROM AUTHOR]