Related papers: Words and characters in finite p-groups

Let G be a finite non-abelian simple group and let p be a prime. We classify all pairs (G,p) such that the sum of the complex irreducible character degrees of G is greater than the index of a Sylow p-subgroup of G. Our classification…

Let G be a finite abelian group. For g in G and i an integer we define N(i,g) to be the number of subsets of G of size i which sum up to g. We will give a short proof, using character theory, of a formula for these N(i,g) due to Li and Wan.…

We continue the study of glider representations of finite groups $G$ with given structure chain of subgroups $e \subset G_1 \subset \ldots \subset G_d = G$. We give a characterization of irreducible gliders of essential length $e \leq d$…

Let $G$ be a finite group of odd order admitting an involutory automorphism $\phi$. We obtain two results bounding the exponent of $[G,\phi]$. Denote by $G_{-\phi}$ the set $\{[g,\phi]\,\vert\, g\in G\}$ and by $G_{\phi}$ the centralizer of…

Let $G$ be a finite group and $d$ the degree of a complex irreducible character of $G$, then write $|G|=d(d+e)$ where $e$ is a nonnegative integer. We prove that $|G|\leq e^4-e^3$ whenever $e>1$. This bound is best possible and improves on…

A group element is called a generalized torsion if a finite product of its conjugates is equal to the identity. We prove that in a nilpotent or FC-group, the generalized torsion elements are all torsion elements. Moreover, we compute the…

Let $n$ be a positive integer and $G(n)$ denote the number of non-isomorphic finite groups of order $n$. It is well-known that $G(n) = 1$ if and only if $(n,\phi(n)) = 1$, where $\phi(n)$ and $(a, b)$ denote the Euler's totient function and…

Let $G$ be a finite group and $\sigma=\{\sigma_{i}|i\in I\}$ be a partition of the set of all primes $\mathbb{P}$, that is, $\mathbb{P}=\bigcup_{i\in I}\sigma_{i}$ and $\sigma_{i}\cap \sigma_{j}=\emptyset$ for all $i\neq j$. A chief factor…

We develop general formulae for the numbers of conjugacy classes and irreducible complex characters of finite p-groups of nilpotency class less than p. This allows us to unify and generalize a number of existing enumerative results, and to…

Let $K:=\mathbb{Q}(G)$ be the number field generated by the complex character values of a finite group $G$. Let $\mathbb{Z}_K$ be the ring of integers of $K$. In this paper we investigate the suborder $\mathbb{Z}[G]$ of $\mathbb{Z}_K$…

In this note we show that for any powerful $p$-group $G$, the subgroup $\Omega_{i}(G^{p^{j}})$ is powerfully nilpotent for all $i,j\geq1$ when $p$ is an odd prime, and $i\geq1$, $j\geq2$ when $p=2$. We provide an example to show why this…

Let $\Gamma$ be a finitely generated nilpotent group and let G be a complex reductive algebraic group. The representation variety $\mathrm{Hom}(\Gamma,G)$ and the character variety $\mathrm{Hom}(\Gamma,G)//G$ each carry a natural topology,…

Given a group G and positive integers k,n, we let B_n=B_n(G) denote the set of all elements x in G such that |x^G|\leq n, and we say that G satisfies the (k,n)-covering condition for commutators if there is a subset S in G such that |S|\leq…

The main purpose of this note is to establish an effective version of the Grunwald--Wang Theorem, which asserts that given a family of local characters $\chi^{v}$ of $K_{v}^{*}$ of exponent $m$ where $v \in S$ for a finite set $S$ of primes…

First, I construct an isomorphism between the categories of (topological) groups of nilpotency class 2 with 2-divisible center and (topological) Lie rings of nilpotency class 2 with 2-divisible center. That isomorphism allows us to…

Consider a group word w in n letters. For a compact group G, w induces a map G^n \rightarrow G$ and thus a pushforward measure {\mu}_w on G from the Haar measure on G^n. We associate to each word w a 2-dimensional cell complex X(w) and…

We present some results on character degree sums in connection with certain characteristics of finite groups such as p-solvability, solvability, supersolvability, and nilpotency. Some of them strengthen known results in the literature.

Let $A$ be a finite nilpotent group acting fixed point freely on the finite (solvable) group $G$ by automorphisms. It is conjectured that the nilpotent length of $G$ is bounded above by $\ell(A)$, the number of primes dividing the order of…

In this note we show that if $p$ is an odd prime and $G$ is a powerful $p$-group with $N\leq G^{p}$ and $N$ normal in $G$, then $N$ is powerfully nilpotent. An analogous result is proved for $p=2$ when $N\leq G^{4}$.

If G is a finitely generated group, and A an algebraic group, then Hom(G,A) is a possibly reducible algebraic variety denoted by R_A(G). Here we define the profile function, P_d(R_A(G)), of the representation variety of G over A to be…