# 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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### Application of character theory to structure of groups

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 ...
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