Let Irr(G) be the set of Irreducible complex characters of a finite group G and cd(G)={x(1) | x Î 2 Irr(G)} be the set of Irreducible character degrees of G. The question: Given the set cd(G), what can be said about the structure of G? have been studied by several people. In case cd(G)={1,m}, for some integer m, the basic tools for studying the question can be found in Chapter 12 of the well known book of Isaacs [8]. Isaacs proved in Theorem 12.5 and Corollary 12.6 the following results. Theorem A. Let G be a finite group and cd(G) = {1,m}. Then, at least one of the followings occurs: (a) G has an abelian normal subgroup of index m.

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Let G be a , nite group, Irr(G) be the set of complex Irreducible characters of G, and denote by cd(G) = f, (1): ,2 Irr(G)g the set of Irreducible character degrees of G. Classifying , nite groups by the properties of their characters is an interesting problem in representation theory. In [3], Huppert conjectured that each , nite non-abelian simple group G is characterized by the set cd(G): In [3, 4, 13, 15], it was shown that the conjecture holds for simple groups such as L2(q) and Sz(q). In this paper, we attempt to characterize the Janko groups J1,J3 and J4 by their orders and one Irreducible character degree. Also authors guess that the result is not correct for Janko group J2, but they have not found any counterexample yet. Let G be a , nite group,L(G) denotes the largest Irreducible character degree of G. The following result is our main theorem in the third section.

Let R be a commutative ring with identity and M be an R-module. In this paper, we will introduce the concept of 2-Irreducible (resp., strongly 2-Irreducible) submodules of M as a generalization of Irreducible (resp., strongly Irreducible) submodules of M and investigated some properties of these classes of modules.

For a character ,of a , nite group G, the number , c(1) = [G: ker, ] , (1) is called the co-degree of , . Let Sol(G) denote the solvable radical of G. In this paper, we show that if G is a , nite non-solvable group with f, c(1)2: ,2 Irr(G)g = f1,2mg for some positive integer m, then G=Sol(G) has a normal subgroup M=Sol(G) such that M=Sol(G) , = PSL2(2n) for some integer n ,2, [G: M] is odd and G=Sol(G). Aut(PSL2(2n)).

In this article, we introduce monomial Irreducible representations of the special linear Lie algebra sln(C). We will show that this kind of representations have bases for which the action of the Chevalley generators of the Lie algebra on the basis elements can be given by a simple formula.