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An obvious choice is the enumeration of orbits of finite group actions, which show up everywhere in middle- and high-school competitions in disguise. The "cute" example here is coloring a cube or a r …

This is a different take on Steven Landsburg's answer. The short version is that conjugacy classes and irreducible representations should be thought of as being dual to each other. Fix an algebraica …

$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Vect{Vect}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Hom{Hom}$The infinitude of the in …

More precisely, is every abelian group a colimit $\text{colim}_{j \in J} F(j)$ over a diagram $F : J \to \text{Ab}$ where each $F(j)$ is isomorphic to $\mathbb{Z}$? Note that this does not follow fro …

Is every (finite-dimensional, complex) representation of a finite group defined over the algebraic integers? Apologies in advance if this is obvious. Edit, 5/31/24: Since this question is getting some …

I actually even struggle to find examples of primitives matrices in these groups. Here is a relatively easy sufficient condition. If $M \in SL_n(\mathbb{Z})$ is the $k^{th}$ power of some other matr …

Acyclic groups must in particular have trivial abelianization, so all of their quotients must be perfect. This is the only obstruction; A.J. Berrick shows in The acyclic group dichotomy (which I just …

A simple bound on the largest dimension of a complex irreducible representation (which is either equal to or half of the largest dimension of a real irreducible representation) is the following: we kn …

There are cool examples already for finite group: Dijkgraaf and Witten recover and generalize a combinatorial formula due to Burnside using the TFT formalism. It's probably worth elaborating on t …

Yes. Some Googling turns up J. M. Tyrer Jones, "Direct products and the Hopf property," J. Austral. Math. Soc. 17 (1974), 174-196.

First let's recall some definitions. Let $G$ be a perfect group, so that $$H^2(G, A) \cong \text{Hom}(H_2(G), A)$$ for all abelian groups $A$ by universal coefficients. This means that when $A = H_ …

Some thoughts. $X$ defines a representation $V = \mathbb{C}^X$ of $G$ with character $\chi(g) = \text{Fix}(g)$, and the projection from $V$ to its invariant subspace is $\frac{1}{|G|} \sum_{g \in G} …

More precisely, what is the smallest exponent e such that, for every n, there exists a group of size at most Cn^e for some absolute constant C and with an n-dimensional irreducible complex representat …

A standard property of Pontrjagin duality is that a locally compact Hausdorff abelian group is discrete iff its dual is compact (and vice versa). In what senses, if any, is this still true for nonabel …

It's cleaner to ask about an arbitrary finite-dimensional real representation $V$ of a finite group $G$; the hypothesis that $V$ is faithful isn't particularly helpful. $V$ has a decomposition $\bigop …