Related papers: Homogeneous products of characters

We study the zero-sharing behavior among irreducible characters of a finite group. For symmetric groups $S_n$, it is proved that, with one exception, any two irreducible characters have at least one common zero. To further explore this…

Let G be a finite group and N be a non-trivial normal subgroup of G, such that the average character degree of irreducible characters in Irr(G|N) is less than or equal to 16=5. Then we prove that N is solvable. Also, we prove the…

A finite group $G$ is called $\psi$-divisible if $\psi(H)|\psi(G)$ for any subgroup $H$ of $G$, where $\psi(H)$ and $\psi(G)$ are the sum of element orders of $H$ and $G$, respectively. In this paper, we extend a result provided in [10], by…

Let $\pi_1,\ldots,\pi_k$ be smooth irreducible representations of $p$-adic general linear groups. We prove that the parabolic induction product $\pi_1\times\cdots\times \pi_k$ has a unique irreducible quotient whose Langlands parameter is…

For a countable group G and a multiplier c on G with values in the circle, we study the property of G having a unitary projective c-representation which is both irreducible and projectively faithful. We show that this property is equivalent…

Denote by $G$ a finite group and let $\psi(G)$ denote the sum of element orders in $G$. In 2009, H.Amiri, S.M.Jafarian Amiri and I.M.Isaacs proved that if $|G|=n$ and $G$ is non-cyclic, then $\psi(G)<\psi(C_n)$, where $C_n$ denotes the…

For a group $G$ and a character $\chi$ of $G$, let $c(\chi)$ denote the set of all irreducible characters of $G$, occurring in $\chi$. We prove that whenever $q\geq 8$, all non-trivial irreducible character $\chi$ of $\mathrm{PSL}_2(q)$…

Let G be any abelian group and {a_sG_s}_{s=1}^k be a finite system of cosets of subgroups G_1,...,G_k. We show that if {a_sG_s}_{s=1}^k covers all the elements of G at least m times with the coset a_tG_t irredundant then [G:G_t]\le 2^{k-m}…

Let $\chi$ be an irreducible character of a finite group $G$. A. R. Miller conjectured that the proportion of elements $g\in G$ such that $\chi(g)$ is zero or a root of unity is at least 1/2. We construct a character of a perfect group of…

The Alperin--McKay conjecture relates irreducible characters of a block of an arbitrary finite group to those of its $p$-local subgroups. A refinement of this conjecture was stated by the author in a previous paper. We prove that this…

The cyclically reduced product of two words is the cyclically reduced form of the concatenation of the two words. While the reduced form of such a concatenation (which is the product of the free group) verifies many basic properties like…

We consider groups where the centers of the irreducible characters form a chain. We obtain two alternate characterizations of these groups, and we obtain some information regarding the structure of these groups. Using our results, we are…

Let $G$ be a finite group and $p\in \pi(G)$, and let Irr$(G)$ be the set of all irreducible complex characters of $G$. Let $\chi \in {\rm Irr}(G)$, we write ${\rm cod}(\chi)=|G:{\rm ker} \chi|/\chi(1)$, and called it the codegree of the…

Let $G$ be a finite group and let $\rm{Irr}(G)$ be the set of all irreducible complex characters of $G$. For a character $\chi \in \rm{Irr}(G)$, the number $\rm{cod}(\chi):=|G:\rm{ker}\chi|/\chi(1)$ is called the co-degree of $\chi$. The…

The character of an irreducible admissible representation of a $p$-adic reductive group is known to be a constant function in some neighborhood of any regular semisimple element $\gamma$ in the group. Under certain mild restrictions on…

In 1968, John Thompson proved that a finite group $G$ is solvable if and only if every $2$-generator subgroup of $G$ is solvable. In this paper, we prove that solvability of a finite group $G$ is guaranteed by a seemingly weaker condition:…

If two homogeneous IFSs satisfying the OSC with opposite common contraction factors share the same attractor on the real line, we show that this attractor is symmetric. This answers a question of Feng and Wang [Adv. Math. 222 (2009),…

Let f: G=* G(i) -> B=* B(i) be a group homomorphism between free products of groups. Suppose that G(i)f=B(i) of all i. Let H be a subgroup of G such that Hf=B. Then H decomposes into a free product H=*H(i) with H(i)f=B(i). Furthermore, H(i)…

R. Stanley has found a nice combinatorial formula for characters of irreducible representations of the symmetric group of rectangular shape. Then, he has given a conjectural generalisation for any shape. Here, we will prove this formula…

A group is called $(m,n)$-bicyclic if it can be expressed as a product of two cyclic subgroups of orders $m$ and $n$, respectively. The classification and characterization of finite bicyclic groups have long been important problems in group…