## Tagged Questions

3answers
253 views

### Characters of p-groups

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 …
1answer
146 views

### Permutation character of the symmetric group on subsets of certain size

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 …
1answer
154 views

### Character table entries and sums of roots of unity

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 …
1answer
356 views

### When does the modulus of a sum of an integer and an algebraic integer equal an integer?

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 …
0answers
194 views

### A problem about primitive Dirichlet characters

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 …
0answers
205 views

### Class function counting solutions of equation in finite group: when is it a virtual character?

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 …
0answers
137 views

### Interplay between two definitions of the transfer homomorphism.

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 …
0answers
308 views

### Degrees of irreducible characters of groups of order 48 [closed]

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
2answers
672 views

### Two Definitions of “Character” of topological groups

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 …
1answer
146 views

### Induced character for non-injective homomorphisms

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_ …
2answers
456 views

### Character group of Frobenius kernels

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 …
2answers
314 views

### Finite, abelian, yet “fugitive” orthogonal subgroups

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\ …
4answers
1k views

### Introduction to L-series and Dirichlet characters?

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 …
2answers
362 views

### When is a class function on a group G (finite abelian) into the rational numbers Q an element of the rational representation ring of G?

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_{\ …
2answers
483 views

### Is 2-sylow subgroup of a rational group also a rational group?

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 …

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