Character covering number of $\mathrm{PSL}_2 (q)$

The degree of an irreducible complex character afforded by a finite group is bounded above by the index of an abelian normal subgroup and by the square root of the index of the center. Whenever a finite group affords an irreducible character whose degree achieves these two upper bounds the group must be solvable.

We consider G to be a finite group and p as a prime number. We fix ψ to be an irreducible character of G with its restriction to all p-regular elements of G and ψ0 to be an irreducible Brauer character. The main aim of this paper is to describe and investigate the relationship between cyclic anchor group of ψ and the defect group of a p-block which contains ψ. Our methods are to study and generalize some facts for the cyclic defect groups of a p-block B to the case of a cyclic anchor group of irreducible characters which belong to B. We establish and prove a criteria for an irreducible character to have a cyclic anchor group.

Irreducible characters and normal subgroups in groups of odd order

Suppose that G is a finite p -solvable group such that N G ( P ) /P has odd order, where P ∈ Syl p ( G ). If χ is an irreducible complex character with degree not divisible by p and field of values contained in a cyclotomic field Q p a , then every subnormal constituent of χ is monomial. Also, the number of such irreducible characters is the number of N G ( P )-orbits on P/P’ . Introduction There are few results guaranteeing that a single irreducible complex character χ ∈ Irr( G ) of a finite group G is monomial. Recall that χ ∈ Irr( G ) is monomial if there is ƛ ∈ Irr( U ) linear such that ƛG = χ . It is known that every irreducible character of a supersolvable group is monomial, for instance, but this result depends more on the structure of the group rather than on the properties of the characters themselves. An exception is a theorem by R. Gow of 1975 ([3]): an odd degree real valued irreducible character of a solvable group is monomial. Recently, we gave in [8] an extension of this theorem which also dealt with the degree and the field of values of the character. (Yet another similar monomiality criterium was given in [9]: if the field of values Q( χ ) of χ is contained in the cyclotomic field Q n and ( χ (1) , 2 n ) = 1, then χ is monomial whenever G is solvable.) In this note, we apply non-trivial Isaacs π -theory of solvable groups to give a shorter proof of the above result at the same time that we gain some new information about the subnormal constituents of the characters, among other things. It does not seem easy at all to prove these new facts without using this deep theory. Recall that for every solvable group and any set of primes π , M. Isaacs defined a canonical subset B π ( G ) of Irr( G ) with remarkable properties ([4]). Since, by definition, every χ ∈ B π ( G ) is induced from a character of π -degree, it is clear that B π -characters of π’ -degree are monomial.

Let G be a finite nontrivial group with an irreducible complex character χ of degree d = χ(1). It is known from the orthogonality relation that the sum of the squares of degrees of irreducible characters of G is equal to the order of G. N. Snyder proved that if |G| = d(d + e), then the order of G is bounded in terms of e, provided e > 1. Y. Berkovich proved that in the case e = 1 the group G is Frobenius with the complement of order d. We study a finite nontrivial group G with an irreducible complex character Θ such that |G| ≤ 2Θ(1)2 and Θ(1) = pq, where p and q are different primes. In this case we prove that G is solvable groups with abelian normal subgroup K of index pq. We use the classification of finite simple groups and prove that the simple nonabelian group whose order is divisible by a prime p and of order less than 2p4 is isomorphic to L2(q), L3(q), U3(q), Sz(8), A7, M11 or J1.

It is well-known [3; V.13.7] that each irreducible complex character of a finite group G is rational valued if and only if for each integer m coprime to the order of G and each g ∈ G, g is conjugate to gm. In particular, for each positive integer n, the symmetric group on n symbols, S(n), has all its irreducible characters rational valued. The situation for projective characters is quite different. In [5], Morris gives tables of the spin characters of S(n) for n ≤ 13 as well as general information about the values of these characters for any symmetric group. It can be seen from these results that in no case are all the spin characters of S(n) rational valued and, indeed, for n ≥ 6 these characters are not even all real valued. In section 2 of this note, we obtain a necessary and sufficient condition for each irreducible character of a group G associated with a 2-cocycle α to be rational valued. A corresponding result for real valued projective characters is discussed in section 3. Section 1 contains preliminary definitions and notation, including the definition of projective characters given in [2].

On the Quadratic Type of Some Simple Self-Dual Modules over Fields of Characteristic Two

In this paper we study finite p-solvable groups having irreducible complex characters chi in Irr(G) which take roots of unity values on the p-singular elements of G.

Let G be a finite nilpotent group, χ and ψ be irreducible complex characters of G with prime degree. Assume that χ(1) = p. Then, either the product χψ is a multiple of an irreducible character or χψ is the linear combination of at least $\frac{p+1}{2}$ distinct irreducible characters.

Persi Diaconis and I. M. Isaacs generalized the character theory to super-character theories for an arbitrary finite group (Diaconis and Isaacs, in Trans Am Math Soc 360(5):2359–2392, 2008). In these theories, the irreducible characters are replaced by certain so-called supercharacters, and the conjugacy classes of the group are replaced by superclasses. Also, Diaconis and Isaacs discussed supercharacter theories and gave some properties of them. We consider in this note certain sums of irreducible Brauer characters and compatible unions of regular conjugacy classes in an arbitrary finite group and we give a generalization of the Brauer character theory to super-Brauer character theories. We also discuss super-Brauer character theories and obtain some results which are similar to those of Diaconis and Isaacs.

Two families of groups close to groups of odd order, and two families of groups close to real groups will be described. The first two are the family of finite groups in which all real irreducible ordinary characters are linear, and the family of all finite groups with the dual condition on the conjugacy classes, namely, groups in which all real conjugacy classes are contained in the center of the group. We will see that each group in one of these families is a direct product of a group of odd order with a 2-group. The families close to real groups are the family of finite groups in which every non-real irreducible ordinary character is linear, and dually, the family of all finite groups in which every non-real conjugacy class is contained in the center. For a group G in the first family we show that the collection of real elements R(G) is a normal subgroupG has a normal π-complement where π = π (∣G: R(G)∣), and a Hall π-subgroup which is either abelian or a nearly real 2-group. A group is in the second family is either abelian, or real or a 2-group. Descriptions and examples of 2-groups satisfying the above conditions will be discussed next. Some of these 2-groups are related to groups treated by several authors. We consider these related groups as well. The interest in the groups mentioned above was triggered by the observation that if G is a finite group in which the square of each irreducible character has at most two irreducible constituents, then all real irreducible characters of G are linear. In the last section of this article we discuss such groups and groups related to them.

Let cd(G) be the set of irreducible complex character degrees of a finite group G. The Taketa problem conjectures that if G is a finite solvable group, then $${{\rm dl}(G) \leqslant |{\rm cd} (G)|}$$ , where dl(G) is the derived length of G. In this note, we show that this inequality holds if either all nonlinear irreducible characters of G have even degrees or all irreducible character degrees are odd. Also, we prove that this inequality holds if all irreducible character degrees have exactly the same prime divisors. Finally, Isaacs and Knutson have conjectured that the Taketa problem might be true in a more general setting. In particular, they conjecture that the inequality $${{\rm dl}(N) \leqslant |{\rm cd} {(G \mid N)}|}$$ holds for all normal solvable subgroups N of a group G. We show that this conjecture holds if $${{\rm cd} {(G \mid N')}}$$ is a set of non-trivial p–powers for some fixed prime p.

Let r > 0 be a prime integer, and let G be a finite general linear group GL(n, q), a unitary group U(n, q) with q = qg, or a symmetric group S(n) of degree n. The partition of the irreducible (ordinary) characters of S(n) into r-blocks of S(n) was given by Brauer's and Robinson's solution of Nakayama's conjecture, see [7], p. 245. In their fundamental paper [5] Fong and Srinivasan have recently classified the r-blocks of GL(n, q) and U(n, q) for all primes r > 2 with (r, q)= 1. Using these classifications of the r-blocks of G we show in this article that there is a natural one-to-one correspondence ~b between the irreducible characters of height zero of an r-block B of G with defect group R and the irreducible characters of height zero of the Brauer correspondent b of B in N=NG(R ) (Theorem (4.10)). In particular, ko(B)=ko(b), where ko(B ) denotes the number of all irreducible characters )~ of B with height h t z = 0 . Therefore Alperin's conjecture on the numbers of irreducible characters of height zero is verified for all general linear, unitary and symmetric groups. If G=S(n), then Theorem(4.10) also holds for r = 2. In order to establish the character correspondence ~, three other correspondences are studied, the product of which is ~b. In Sect. 1 we construct for every r-block B of G with defect group R a subgroup G of G with a Sylow r-subgroup /~-~R such that there is a natural height preserving one-to-one correspondence T between the set Irr(B) of all irreducible characters of B and the set Irr(/~0) of all irreducible characters of the principal r-block/~o of d (Reduction Theorems (1.9) and (1.10)). The map ~ respects the geometric conjugacy classes of characters. If /~0 denotes the principal r-block of N=Nd(/~), and if b is the Brauer correspondent of B in N=N6(R), then by Theorems (3.8) and (3.10) the block ideals/~o and b are Morita equivalent, have the same decomposition numbers, and there is a natural height preserving one-to-one correspondence a between Irr(bo) and Irr(b).

Principal blocks with 5 irreducible characters

For a prime p, we determine a Sylow p-subgroup D of a finite group G such that the principal p-block B of G has four irreducible ordinary characters. It has been determined already for the cases where the number is up to three by work by R. Brauer, J. Brandt, and V.A. Belonogov 30 years ago. Our proof relies on the classification of finite simple groups.