# Tag Info

### How to find normal subgroups from a character table?

This is quite well-known and can be found in books on representation theory. Here is an explanation, which is far from being original. First fact : $N$ is a normal subgroup of a finite group $G$ if ...

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### How to construct the $C_{4v}$ character table given its Cayley table

OK, so this author has some really wonky notation, but it's all fine in the end. The first comment is that the pyramid is a red herring. All symmetries of the pyramid leave the peak fixed, so actually ...
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### Perfect groups whose character degrees square divide its order

If you extend to perfect groups, this is pretty easy. Let $G$ be your favourite simple group, and let $A$ be a very large abelian group. A semidirect product $X=A\rtimes G$ has character degrees ...
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Suppose that $G=C_2$ is cyclic of order 2 with generator $x$. Define $\chi$ by $\chi(1)=1$, $\chi(x)=-1$. A reasonable choice for the square root would be $\sqrt{1}=1$, $\sqrt{-1}=i$, but then $$\... 6 votes Accepted ### a formula involving order of Dirichlet characters, \mu(n) and \varphi(n) Fix n and define$$f(d)=\sum_{o(\chi)=d}\chi(n).$$Let's show f is multiplicative. First off, let g be a generator for (\mathbb{Z}/p\mathbb{Z})^\times and write n=g^k, then let \psi be a ... 6 votes Accepted ### sum of roots of unity is algebraic integer Let us assume that all the \alpha_j are m-th roots of unity. So if we fix some primitive m-th root of unity \zeta then \beta=\alpha_1+\cdots+\alpha_k=\zeta^{a_1}+\cdots+\zeta^{a_n} for some ... 6 votes Accepted ### characters of finite groups which vanish except identity This character \chi is a multiple of the character of the regular representation: \chi=a\chi_{reg}. A priori a\in\Bbb Q, but the regular representation contains one copy of the trivial ... 6 votes Accepted ### The condition between \chi(1) and [G:H] which gives us a normal subgroup. Any reference here is to the book of Marty Isaacs, Character Theory of Finite Groups. Now for (b) you must be a bit more precise: if H is abelian, and \chi is an irreducible character of G, then ... 6 votes Accepted ### Monster coefficients In GAP, you could simply iterate in a triple loop over \phi,\psi,\rho, calculate the g-values and find maximum and sum values: ... 6 votes Accepted ### How does knowing that \sqrt{7}\notin\mathbb{Q}(e^{2\pi i/7}) help construct this character table? Okay, I finally worked it out. I'll give a complete solution to the exercise: First of all we have the trivial character, which we'll denote \chi_1. We have \left<\chi_1,\alpha\right>=\left&... 6 votes ### Find all irreducible characters of a matrix group on finite field \mathbb F_5 (Partial answer / hint, starting with character degrees) I am assuming that you are taking about representations over \mathbb{C}. As indicated G=HN, N \lhd G, H \cap N=1 with H \cong C_4 and ... 6 votes ### Validating a Character Table for a Given Finite Group In general, it is not easy to decide if a given matrix is a character table. For instance, the matrix$$\begin{pmatrix} 1&1&1&1\\ 1&1&-1&1\\ 2&2&0&-1\\ 6&-1&...
If you do not wish to use character theory or calculations in a group ring for that matter, try this one: for a finite group $G$, let $g \in G$ be a commutator (that is, $g=[x,y]=x^{-1}y^{-1}xy$ for ...