Let $p$ a prime number, ${q_{_1}}$,…, ${q_{_r}}$ are the distinct primes dividing $p-1$, ${\mu}$ is the Möbius function, ${\varphi}$ is Euler’s phi function, ${\chi}$ is Dirichlet character $\bmod{p}$ and ${o(\chi)}$ is the order of ${\chi}$. How can I show that: $$\sum\limits_{d|p – 1} {\frac{{\mu (d)}}{{\varphi (d)}}} \sum\limits_{o(\chi ) = d} {\chi (n)} = \prod\limits_{j = […]

There is a well-known result of Solomon which states that sum of entries of any row in $\mathbb{C}$-character table of a group $G$ is an integer number. It is mentioned in Martin Isaacs Character Theory of Finite Group as a note that the column sums are also integers. My question is that what’s the reason […]

I wanted to find all representations of the group $\Bbb C^\times$ under multiplication. I was thinking that I could have a degree $n$-representation of this for any $n\in \Bbb N$ by letting $z\in \Bbb C^\times$ have matrix $\rho_z=zI_{n\times n}$ which acts on $\Bbb C^n$ and is a group homomorphism. Then I was wondering if I […]

Sorry for any mistakes I make here, this is my first post here. I have a group $G$ which has an abelian subgroup $A<G$. I also know there is a irreducible character $\chi$ with the degree of $\chi$ equal to the index of $A$ in $G$. This implies $G$ has a non-identity abelian NORMAL subgroup? […]

Let $n\in\mathbb N$ be a natural number and $\lambda=(a,b)\vdash n$ a partition of $n$ into two parts, i.e. $a\ge b$ and $a+b=n$. In this special case, is there a simple description of the character $\chi_\lambda$ of the irreducible $S_n$-representation corresponding to $\lambda$? I have tried to deduce something from the Frobenius character formula and also […]

Most questions that I can find on here (or anywhere else on the internet) deal with constructing a character table given a description of the group. I’m trying to answer a question which goes the other way though: Given the following character table, where $\alpha=(-1+i\sqrt{7})/2$, what is the order of each $g_i$? I’ve had a […]

The values of irreducible characters of a finite groups are always sums of roots of unity; do all sums of roots of unity (i.e. algebraic integers in the maximal abelian extension of $\mathbb{Q}$) actually appear among the character tables of finite groups? Obviously roots of unity themselves appear, as can be seen from the character […]

Here are the statements of the two theorems. The first statement I took from a paper I have been reading, but I believe can also be found in Isaacs’ Character Theory of Finite Groups as an exercise. The second comes directly from Isaacs’ Finite Group Theory. For both statements, let $X$ be a set of […]

Is there some natural or character-theoretic description of the minimum value of d such that G has a regular orbit on Ωd, where G is a finite group acting faithfully on a set Ω? Motivation: In some cases, there are so many kinds of orbits of G on Ω that every subgroup of G appears […]

For anyone familiar with Artin’s Algebra book, I just worked through the proof of the following theorem, which can be seen here: (5.9) Theorem Let $G$ be a group of order $N$, let $\rho_1,\rho_2,\dots$ represent the distinct isomorphism classes of irreducible representations of $G$ and let $\chi_i$ be the character of $\rho_i$. (a) Orthogonality relations: […]

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