# Irreducible Character

## Top PDF Irreducible Character: ### Bipartite divisor graph for the set of irreducible character degrees

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 . 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  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 , we conclude that mh ∈ cd(G) ∗ , so l = mh, cd(G) = { 1, p α , q β , p α q β } and ### A new characterization of Mathieu groups by the order and one irreducible character degree

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. ### Detecting the prime divisors of the character degrees and the class sizes by a subgroup generated with few elements

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. ### Total characters and Chebyshev polynomials

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. ### Characterization of some simple \$K_4\$-groups by some irreducible complex character degrees

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 , 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 ,  and , it is shown that L 2 (p), simple K 3 -groups and Mathieu simple groups are determined uniquely by their ### Anchors of irreducible characters

character of N . By construction, neither ˜ α nor ˜ β is G-stable. Hence, also by general results on p-factorable characters, ˜ α β ˜ is not G-stable. Since |G/N | = p, it follows that χ is an irreducible character of G. Now, since ˜ β is not G-stable, it is easy to see that χ is not p-factorable. On the other hand, N is a maximal normal subgroup of G. Thus (N, α ˜ β) is a nucleus of ˜ G in the sense of , and the Sylow p-subgroups of N are the first components of the Navarro vertices of χ. ### On nonsolvable groups whose prime degree graphs have four vertices and one triangle

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. ### Utility of irreducible group representations in differential equations

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: ### Class two p groups as fixed point free automorphism groups

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] ### Singh

Fourier matrices are a fundamental ingredient of modular data. Modular data is a basic component of rational conformal field theory, see . Further, rational conformal field theory has important applications in physics, see  and . In particular, it has nice applications to string theory, statistical mechanics, and condensed matter physics, see  and . Modular data give rise to fusion rings, C-algebras and C ∗ -algebras, see  and . 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 . 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 . 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. ### Applications and methods of group theory in elementary particle physics

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] ### On the irreducible characters of the Weyl groups

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] ### Primitive irreducible linear groups

(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 . ### A note on irreducible Heegaard diagrams

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. ### ANAPHYLACTOID PURPURA COMPLICATED BY AN IRREDUCIBLE INTUSSUSCEPTION

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] ### Fully idempotent near rings and sheaf representations

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] ### The irreducible subgroups of exceptional algebraic groups

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 ### Irreducible Representations of Groups of Order 8

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 ): ### Interval Valued Fuzzy Prime Bi-ideals and Interval Valued Fuzzy Strongly Prime Bi-ideals in Near-rings

The notion of bi-ideals in near-rings was brought forth by Chelvam and Ganesan . Abbassi and Rizvi  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 ### Breaking the Hidden Irreducible Polynomials Scheme

In this paper we have shown a full break of the Hidden Irreducible Polynomials scheme introduced by Gómez in . 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.