Let $G$ be an (additive) abelian group. Is there a concise, standard term for the order of the kernel $K$ of the endomorphism $g\mapsto 2g$? Or alternatively, for the index of $K …

Geometric group theory is mainly concerned with topological and geometric properties of groups, spaces on which they act etc., so the ideas employed in GGT are mainly algebraic/geo …

Let $G$ be a finite group. Do eigenvalues of its Cayley graph say anything about the algebraic properties of $G$? The spectrum of Cayley graph may depend on the presentation, so it …

The following result was mentionned earlier in this thread, I searched a bit in the related threads and couldn't find a proof. I would really like to see a proof of it: Let $G$ be …

Related to an answer to a previous question. The answer assume the following result: Let $G$ be a finite group and $\rho : G \rightarrow \text{GL}(\mathbb{C}, n)$ be a faithful re …

Consider the number of fixed points in a permutation chosen uniformly at random from the symmetric group on $n$ elements - this gives a probability distribution. For $k < n$, t …

Is there a systematic way to solve equations in the braid groups? In particular, if B3 is the braid group on three strands with the presentation { a,b | aba = bab }, how do I find …

I'm merely a grad student right now, but I don't think an exploration of the sporadic groups is standard fare for graduate algebra, so I'd like to ask the experts on MO. I did a li …

This question is related to this one: Continued fractions using all natural integers. Suppose we have the set of natural numbers $N$ with order and we perform permutation on it. So …

Consider a monoidal category C with operation $\otimes$, unit object $1$, and diagonal map $\delta:A \to A \otimes A$ for all $A \in C$ (with naturality conditions on the diagonal …

Given a group of order $n$ where $n$ is either a specific number, or a number of a particular form, e.g. square-free, when does $n$ completely determine a particular group property …

To be more precise (but less snappy): is there an example of a group G with finitely many infinite-index subgroups H_1, ..., H_n and elements k_1, ..., k_n such that G is the union …

Let $G$ be a (complex algebraic) group, $H$ a subgroup, and ${\cal O}(G)$ and ${\cal O}(H)$ the coordinate algebras of complex regular functions of $G$ and $H$ respectively. Can ${ …

Let $\rho : S_n \rightarrow \text{GL}(n, \mathbb{C})$ be the homomorphism mapping a permutation $g$ to its permutation matrix. Let $\chi(g) = \text{Trace}(\rho(g))$. What is the v …

Almost 25 years ago a professor at Indiana U showed me the following problem: given a map $\mathbb{Z}^2\rightarrow\mathbb{R}$ such that the sum inside every square (parallel to th …