which elements of a finite group are non-vanishing?

Which elements of a finite group are non-vanishing?

‎Let $G$ be a finite group‎. ‎An element $gin G$ is called non-vanishing‎, ‎if for‎ ‎every irreducible complex character $chi$ of $G$‎, ‎$chi(g)neq 0$‎. ‎The bi-Cayley graph ${rm BCay}(G,T)$ of $G$ with respect to a subset $Tsubseteq G$‎, ‎is an undirected graph with‎ ‎vertex set $Gtimes{1,2}$ and edge set ${{(x,1),(tx,2)}mid xin G‎, ‎ tin T}$‎. ‎Let ${rm nv}(G)$ be the set‎ ‎of all non-vanishi...

group actions related to non-vanishing elements

‎we characterize those groups $g$ and vector spaces $v$ such that $v$ is a faithful irreducible $g$-module and such that each $v$ in $v$ is centralized by a $g$-conjugate of a fixed non-identity element of the fitting subgroup $f(g)$ of $g$‎. ‎we also determine those $v$ and $g$ for which $v$ is a faithful quasi-primitive $g$-module and $f(g)$ has no regular orbit‎. ‎we do use these to show in ...

Rings in which elements are the sum of an‎ ‎idempotent and a regular element

Let R be an associative ring with unity. An element a in R is said to be r-clean if a = e+r, where e is an idempotent and r is a regular (von Neumann) element in R. If every element of R is r-clean, then R is called an r-clean ring. In this paper, we prove that the concepts of clean ring and r-clean ring are equivalent for abelian rings. Further we prove that if 0 and 1 are the only idempotents...

Finite Groups With a Certain Number of Elements Pairwise Generating a Non-Nilpotent Subgroup

Generating Random Elements of a Finite Group

We present a “practical” algorithm to construct random elements of a finite group. We analyse its theoretical behaviour and prove that asymptotically it produces uniformly distributed tuples of elements. We discuss tests to assess its effectiveness and use these to decide when its results are acceptable for some matrix groups.

The probability that a pair of elements of a finite group are conjugate

Let G be a finite group, and let κ(G) be the probability that elements g, h ∈ G are conjugate, when g and h are chosen independently and uniformly at random. The paper classifies those groups G such that κ(G) ≥ 1/4, and shows that G is abelian whenever κ(G)|G| < 7/4. It is also shown that κ(G)|G| depends only on the isoclinism class of G. Specialising to the symmetric group Sn, the paper shows ...