# Questions tagged [gr.group-theory]

Questions about the branch of algebra that deals with groups.

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### Equivalence of category of internal groups and the category of groups

Is the category of internal groups in set equivalent to the category of groups or isomorphic to it or are they just equal? When we define an internal group in Set since the product is unique only up ...
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### Reversals of autonomous subsets in right-angled Coxeter groups

This question has to do with some experimental phenomenon in groups generated by involutions that I cannot explain. Let $G$ be a finite, undirected graph, and let $W$ be the corresponding right-angled ...
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### The principal congruence subgroup of the symplectic group over the integers

Consider the symplectic group $\text{Sp}_{2g}(\mathbb{Z})$ over the integers. It has a classical root system $C_g$ and associated root subgroups $U_\varphi$ for $\varphi\in C_g$. These subgroups are ...
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### Reference for the Brauer-Nesbitt theorem

In Wikipedia's article on the Brauer-Nesbitt theorem, they state that given a group $G$ and a field $E$, two semisimple representations $\rho_1,\rho_2 : G\longrightarrow \operatorname{GL}_n(E)$ are ...
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### Set of $U(6)$ elements which leave a Lie-algebra element invariant under conjugation

Consider the specific element of the corresponding Lie algebra $\mathbb{1}_3 \times \sigma^3$, where $\mathbb{1}_3$ is the unit matrix in 3 dimensions, $\sigma^3$ is the 3rd Pauli matrix and $\times$ ...
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### Good algorithmic properties for quotients of braid groups

I'm trying to understand some things about quotients of braid groups, and particularly I'd like to solve the word problem for some elements of these quotients. I'm using MAGMA to try to access this, ...
Let $\Gamma$ a Fuchsian group acting on the hyperbolic plane $\mathfrak{H}$. For me, I am most interested in the case where $\Gamma$ has a fundamental domain that is a finite-polygon with all ...