Berkovich mentioned the following result of Mann in his book on p-groups: The number of nonlinear irreducible characters of given degree in a p-group is divided by p-1. Do you kn …

The symmetric group $S_n$ acts on $[n]:=\{1,\ldots,n\}$, thereby inducing an action on the set $$\wp_k(n)=\{\: A\subseteq[n] \::\: \#A=k \:\}$$ of subsets of cardinality $k$, simpl …

It is well-known that the entries of the character table of a finite group are sums of roots of unity. Question: Is the converse true? Explicitly, given $z\in \mathbb{Z}[\mu_\inf …

Let say Z is a sum of n-roots of unity and thus an algebraic integer, and D is an rational integer. If |z+D| is an integer, what can we conclude regarding Z? can we say |Z| is in …

Show that for a primitive character $\chi\bmod q$, $q\gt1$ $\sum_{|n|\le q}(1-|n|/q)\chi(n)e(na)\ll\sqrt q$ uniformly in $a$, where $e(na)=\exp(2\pi ina)$ Thank you for any co …

Let $w=w(x_1,\dots,x_n)$ be a word in a free group of rank $n$. Let $G$ be a finite group. Then we may define a class function $f=f_w$ of $G$ by $$ f_w(g) = |\{ (x_1,\dots, x_n)\in …

The transfer homomorphism can be defined in a couple of ways. For the purposes of this question, assume that all groups are finite. Defintion 1. Let $1\leqslant K'\leqslant L …

Hello i wonder if there is a simple argument to show that no group of order 48 can have an irreducible character of degree larger than 4? Thanks, Karim

When I first met the concept of "characters" of topological groups in Pontryagin's book "Topological groups", it was defined as follows: Let $G$ be a topological group. A characte …

Any group homomorphism $\phi\colon H\to G$ gives rise to an induction/restriction adjunction between $G$-representations and $H$-representations: $$ \hom_G(\phi_! M, N) \cong \hom_ …

Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of characteristic $p$ (e.g., $G=SL_n(k)$). Then $G$ is equal to its derived subgroup $[G,G]$. Consequ …

A popular concept in quantum computation, used extensively to design algorithms for finite-abelian-groups, are the so-called orthogonal subgroups Let $G=\mathbb{Z}_{d_1}\times\ …

I'm looking for an introductory text on Dirichlet characters and the L-series of a field K, specifically for quartic extensions of $\mathbb{Q}$. I have Davenport's Multiplicative …

Given a class function $f: G \to \mathbb Q$, where $G$ is a finite abelian group, is there an easy way to decide whether $f$ is an element of the rational representation ring $R_{\ …

As we know, a finite group $G$ is a rational group if $\chi (g)\in\mathbb{Q}$, where $\chi$ is every irreducible charahter and $g\in G$. I have an interesting question that is "Is …