F (H)], [H : F (H)] } . As ∆(H) has two connected components, we can see that H is either a group of type one or four in the sense of [8]. Also by [17, Lemma 5.1] we deduce that m = [G : F (G)] and cd(F (G)) = {1, h q ′ }. Thus case (iv) occurs with s = q. Now suppose B(G) : m − q − l − p − h, where p and q are distinct prime numbers. Suppose K G is a Frobenius group with abelian Frobenius complement of order f and Frobenius kernel N K = ( K G ) ′ which is an elementary abelian s-group for some prime s. Thus N G is abelian. If f = m and s / ∈ ρ(G), then by Gallagher’s theorem [5] we conclude that cd(G) = {1, p α , q β , p α q β }, so case (iii) occurs. If f = m and s ∈ ρ(G), then s = p. Let θ ∈ Irr(N ) be nonlinear. If s does not divide θ(1), then [G : N ]θ(1) ∈ cd(G). Thus l = mθ(1). As (s, f ) = 1, s divides θ(1) which is a contradiction. Thus s divides each nonlinear **irreducible** **character** degree of N , so N has a normal s-complement. Hence we have case (ii). Let f = l. As (s, f ) = 1, s / ∈ ρ(G). On the other hand, for ψ ∈ Irr(N ) with ψ(1) ̸ = 1, [G : N ]ψ(1) ∈ / cd(G). [5, Theorem 12.4] implies that s | ψ(1) and s ∈ ρ(G) which is a contradiction. So f ̸ = l. Finally, suppose K G is an s-group for a prime s. By symmetry we may assume that s = p and f = p α = h. Let χ ∈ Irr(G) with χ(1) = m. By Gallagher’s theorem [5], we conclude that mh ∈ cd(G) ∗ , so l = mh, cd(G) = { 1, p α , q β , p α q β } and

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the conjecture. In this paper, we manage to characterize the ﬁnite simple groups by less **character** quantity. Let G be a ﬁnite group; L(G) denotes the largest **irreducible** **character** degree of G and S(G) denotes the second largest **irreducible** **character** degree of G. We characterize the ﬁve Mathieu groups G by the order of G and its largest and second largest **irreducible** **character** degrees. Our main results are the following theorems.

Abstract. We prove that every finite group G contains a three-generated subgroup H with the following property: a prime p divides the degree of an **irreducible** **character** of G if and only if it divides the degree of an **irreducible** **character** of H. There is no analogous result for the prime divisors of the sizes of the conjugacy classes.

The total **character** τ of a ﬁnite group G is deﬁned as the sum of all the **irreducible** characters of G. K. W. Johnson asks when it is possible to express τ as a polyno- mial with integer coeﬃcients in a single **irreducible** **character**. In this paper, we give a complete answer to Johnson’s question for all ﬁnite dihedral groups. In particular, we show that, when such a polynomial exists, it is unique and it is the sum of certain Chebyshev polynomials of the ﬁrst kind in any faithful **irreducible** **character** of the dihedral group G.

Throughout this paper, let G be a finite group and let all characters be complex characters. Also, let l(G) be the largest **irreducible** **character** degree of G, s(G) be the second largest **irreducible** **character** degree of G and t(G) be the third largest **irreducible** **character** degree of G. The set of all **irreducible** characters of G is shown by Irr(G) and the set of all **irreducible** **character** degrees of G is shown by cd(G). In [4], B. Huppert conjectured that if G is a finite group and S is a finite non-abelian simple group such that cd(G) = cd(S), then G ∼ = S × A, where A is an abelian group. In [7], [11] and [12], it is shown that L 2 (p), simple K 3 -groups and Mathieu simple groups are determined uniquely by their

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Abstract. Let G be a finite group. The prime degree graph of G, denoted by ∆(G), is an undirected graph whose vertex set is ρ(G) and there is an edge between two distinct primes p and q if and only if pq divides some **irreducible** **character** degree of G. In general, it seems that the prime graphs contain many edges and thus they should have many triangles, so one of the cases that would be interesting is to consider those finite groups whose prime degree graphs have a small number of triangles. In this paper we consider the case where for a nonsolvable group G, ∆(G) is a connected graph which has only one triangle and four vertices.

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Now one easily sees that writing a representation given by as a direct sum of **irreducible** ones is equivalent to diagonalizing . So a representation is completely reducible iff is diagonalizable. Since not every linear operator is diagonalizable, not every representation is completely reducible. Thus, more modest goals of the representation theory would be answering the following questions:

irreducible representations are cyclic or the central product of an extra special p group with a cycl i c group of.. That is, all irreducible r epresentations.[r]

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Fourier matrices are a fundamental ingredient of modular data. Modular data is a basic component of rational conformal field theory, see [5]. Further, rational conformal field theory has important applications in physics, see [4] and [8]. In particular, it has nice applications to string theory, statistical mechanics, and condensed matter physics, see [10] and [13]. Modular data give rise to fusion rings, C-algebras and C ∗ -algebras, see [3] and [11]. These rings and algebras are interesting topics of study in their own right. A unitary and symmetric matrix whose first column has positive real entries is called a Fourier matrix if its columns under entrywise multiplication produce integral structure constants. The set of columns of a Fourier matrix under entrywise multiplication and usual addition generate a fusion algebra, see [3]. But a two-step rescaling on Fourier matrices gives rise to self-dual C-algebras. Cuntz, using a computer, classified the Fourier matrices with rational entries up to rank 12, see [3]. But rational Fourier matrices do not include some other important matrices, see sections 4, 5 and 6. Here we use C-algebra perspective to classify the complex Fourier matrices up to rank 5 under certain conditions. Also, we establish some results that are helpful in recognizing the C-algebras that are not arising from Fourier matrices by mere looking at the first row of their **character** tables.

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The use of Schur function methods in the evaluation of Kronecker products of irreducible representations of the compact semisimple Lie groups is reviewed... irreducible representations o[r]

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for Weyl groups of type B, similar to ones in types C and D §'.3 and §4.2, and include some results on inducing up irreducible characters from maximal Weyl subgroups of this group... WB [r]

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(ii) By Huppert 1967, page 469, every algebra with a faithful **irreducible** module is simple, and by the well- known Wedderburn*s theorem (Huppert 1967* page 472), H j is thus isomorphic to a full kxk matrix ring over some skewfield. It is not hard to see that if H is Abelian, the skew field has to be commutative and k = 1 .

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We construct a Heegaard diagram of genus three for the real projective 3-space, which has no waves and pairs of complementary handles. The first example was given by Im and Kim but our diagram has smaller complexity. Furthermore the proof presented here is quite di ﬀ erent to that of the quoted authors, and permits also to obtain a simple alterna- tive proof of their result. Examples of **irreducible** Heegaard diagrams of certain connected sums complete the paper.

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Of the 14 patients with intussusception corn- plicating purpura in Bailey’s review, nine were operated upon with a mortality of 11%, while for the five not operated upon, the mortality w[r]

The next proposition shows that the concepts of prime, irreducible and strongly irreducible coincide for fully idempotent near-rings.. First we record the following existence result for [r]

to make it easier for the reader to follow the tables, and is explained below. We need to describe what the entries in each column of a generic row represent. If the first column is empty then this row is a heading of a section or part, as described above. There are two other types of rows, both of which correspond to G-**irreducible** subgroups X contained in Y , where Y is the subgroup of the current section or part, contained in the maximal connected subgroup M (in this description we are including the case Y = M ). In both cases, the entry in the first column gives the ID number for the conjugacy class or classes of such **irreducible** subgroups. The second and third column differ, depending on whether X is a diagonal subgroup of Y or not. If X is a diagonal subgroup of Y then the second and third columns are merged and the information given is of the form “X via . . . ”, denoting the usual notation for an embedding of a diagonal subgroup of Y , as well as any restrictions on the characteristic p. If X is not a diagonal subgroup of Y then the second column gives the isomorphism type of X and any restrictions on the characteristic p. The third column contains the description of V M ↓ X . Note

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There is a specific method to find the **irreducible** representations for cyclic groups using this method. A cyclic group is Abelian and each of its h elements is in a separate class. It also has h 1-dimensional **irreducible** representations. In order to obtain the **irreducible** representations for a cyclic group, the exponential below is used as the pth **irreducible** representation, Γ p (C n ):

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The notion of bi-ideals in near-rings was brought forth by Chelvam and Ganesan [8]. Abbassi and Rizvi [1] meditated and research upon the prime ideals in near-rings in 2008. In this discourse, we have thoroughly studied i-v fuzzy prime bi-ideals, i-v fuzzy semiprime bi-ideals, i-v fuzzy reducible and i-v fuzzy **irreducible** bi-ideals in near-rings. 2. Preliminaries

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In this paper we have shown a full break of the Hidden **Irreducible** Polynomials scheme introduced by Gómez in [1]. We have shown that the private key is publicly known by the design of the system. Moreover, we have shown that due to the construction of the private map, namely univariate polynomial multiplication, one can even easily read off the transformation matrix for the system of multivariate quadratic polynomial equations such that not even linear algebra is needed for attacking the scheme.