Groups whose set of vanishing elements is exactly a conjugacy class

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 G be a finite group, and g ∈ G . Then g is said to be a vanishing element of G, if there exists an irreducible character χ of G such that χ ( g ) = 0 . Denote by Vo ( G ) the set of the orders of vanishing elements of G. We say a non-abelian group G is V-recognizable, if any group N with Vo ( N ) = Vo ( G ) is isomorphic to G. In this paper, we investigate the V-recognizability of E 8 ( p ) , where p is a prime number. As an application, among the 610 primes p with p < 10000 and p ≡ 0 , 1 , 4 ( mod 5 ) , we obtain that the method is always valid for confirming the V-recognizability of E 8 ( p ) for all such p but 919, 1289, 1931, 3911, 4691, 5381 and 7589.

For a finite group 𝐺, an element is called a vanishing element of 𝐺 if it is a zero of an irreducible character of 𝐺; otherwise, it is called a non-vanishing element. Moreover, the conjugacy class of an element is called a vanishing class if that element is a vanishing element. In this paper, we describe finite groups whose vanishing class sizes are all prime powers, and on the other hand we show that non-vanishing elements of such a group lie in the Fitting subgroup which is a proof of a conjecture mentioned in [I. M. Isaacs, G. Navarro and T. R. Wolf, Finite group elements where no irreducible character vanishes, J. Algebra 222 (1999), 2, 413–423] under this special restriction on vanishing class sizes.

We say that a finite group $G$ is conjugacy expansive if for any normal subset $S$ and any conjugacy class $C$ of $G$ the normal set $SC$ consists of at least as many conjugacy classes of $G$ as $S$ does. Halasi, Mar'oti, Sidki, Bezerra have shown that a group is conjugacy expansive if and only if it is a direct product of conjugacy expansive simple or abelian groups. By considering a character analogue of the above, we say that a finite group $G$ is character expansive if for any complex character $alpha$ and irreducible character $chi$ of $G$ the character $alpha chi$ has at least as many irreducible constituents, counting without multiplicity, as $alpha$ does. In this paper we take some initial steps in determining character expansive groups.

Introduction. In recent years, there have been interests in studying the influence of arithmetic conditions on the lengths of conjugacy classes to the structure of finite groups. The results are parallel to those of character degrees, and lead to strong restrictions to the group structure. An interesting relevant topic is the lengths of conjugacy classes of subgroups. This was first brought up to the author by Professor J. P. Zhang, who has obtained some interesting results on conjugacy class lengths of Sylow subgroups, the so called Sylow numbers [91. In this paper, we consider conjugacy class lengths of cyclic subgroups of finite goups. Our notations are standard and tends to follow [5]. All groups considered are finite. Let G be a finite group, H a subgroup of G, then the length of conjugacy class of H, the number of subgroups of G conjugate to H, is equal to I G : NG(H)[. Our main result is Theorem 1, its proof depends on the classification of finite simple groups.

Let G be a finite group. Define a relation ~ on the conjugacy classes of G by setting C ~ D if there are representatives c ∈ C and d ∈ D such that cd = dc. In the case where G has a normal subgroup H such that G/H is cyclic, two theorems are proved concerning the distribution, between cosets of H, of pairs of conjugacy classes of G related by ~. One of the proofs involves an application of the famous marriage theorem of Philip Hall.

Let G be a finite group and let p be a prime. In this paper, we classify all finite quasisimple groups in which the degrees of all irreducible p-Brauer characters are prime powers. As an application, for a fixed odd prime p, we classify all finite nonsolvable groups with the above-mentioned property and having no nontrivial normal p-subgroups. Furthermore, for an arbitrary prime p, a complete classification of finite groups in which the degrees of all nonlinear irreducible p-Brauer characters are primes is also obtained.

We show that, for any positive integer k, there are only finitely many finite groups, up to isomorphism, with exactly k conjugacy classes of elements of prime power order. This generalizes a result of E. Landau from 1903. The proof of our generalization makes use of the classification of finite simple groups.

Perfect nonlinear functions between two finite abelian groups were studied by Carlet and Ding (J Complex 20:205---244, 2004) and Pott (Discret Math Appl 138:177---193, 2004), which can be regarded as a generalization of bent functions on finite abelian groups studied by Logachev et al. (Discret Math Appl 7:547---564, 1997). Poinsot (J Discret Math Sci Cryptogr 9:349---364, 2006), (Cryptogr Commun 4:1---23, 2012) extended this research to arbitrary finite groups, and characterized bent functions on finite nonabelian groups as well as perfect nonlinear functions between two arbitrary finite groups by the Fourier transforms of the related functions at irreducible unitary representations. The purpose of this paper is to study the characterizations of the bentness (perfect nonlinearity) of functions on arbitrary finite groups by the Fourier transforms of the related functions at irreducible characters. We will also give a characterization of a perfect nonlinear function by the relative pseudo-difference family.

In the general case, the order of a finite nonidentity group G is substantially larger than the squared degree of every irreducible character Θ of G, i.e., Θ(1)2< G. In the present paper, we study finite groups with an irreducible character Θ such that $$\left| G \right| \leqslant 2\Theta {\left( 1 \right)^2}$$ .

In his paper Finite groups have many conjugacy classes (J. London Math. Soc (2) 46 (1992), 239–249), L. Pyber proved the to-date best general lower bounds for the number of conjugacy classes of a finite group in terms of the order of the group. In this paper we strengthen the main results in Pyber’s paper.

Simple classical groups of Lie type are determined by their character degrees

Variants of some of the Brauer-Fowler theorems

Let G be a finite group and let w A IrrðGÞ lie in a p-block of full defect for an odd prime p. We investigate mQðwÞp, the p-part of the rational Schur index of w. At the onset of this work, it was suspected that these Schur indices would not be divisible by p, and in Section 2 below, we show this to be the case when G is either supersolvable or p-nilpotent. Further motivation came from the situation in which G has a normal Sylow p-subgroup. Of course in this case all irreducible characters of G lie in p-blocks of full defect. Recall that a p-hyperelementary subgroup is a subgroup with a cyclic, normal p-complement and that by results of Brauer, Witt, and Berman G has a p-hyperelementary subgroup H with z A IrrðHÞ such that mQðwÞp divides mQðzÞ. If the Sylow p-subgroup of G is normal, all p-hyperelementary subgroups of G are nilpotent. Hence, the Schur indices of their irreducible characters are at most 2, and it follows that p does not divide mQðwÞ: One final case is that in which the Sylow p-subgroups of G are abelian. In this situation, Brauer’s height zero conjecture implies that the degrees of the irreducible characters in p-blocks of full defect are not divisible by p. Thus their Schur indices are not divisible by p either. However, in Section 3 we fix an a A N and construct a solvable group G with irreducible character w lying in the principal p-block of G=KerðwÞ and satisfying mQðwÞ 1⁄4 p. It is crucial to regard w as a faithful irreducible character of G=KerðwÞ, as one can always regard an irreducible character c of a group H as an irreducible character of a larger group G which surjects onto H. This does not a¤ect the Schur index of c, but it can easily be set up to ensure c lies in the principal p-block of G: The group SLð2; 3Þ has a rational-valued irreducible character w such that

Let G be a finite group and a ∈G. Let a G ={g −1 a g | g ∈ G} be the conjugacy class of a in G. Assume that a G and b G are conjugacy classes of G with the property that C G (a)=C G (b). Then a G b G is a conjugacy class if and only if [a,G]=[b,G]=[ab,G] and [ab,G] is a normal subgroup of G.