📘 group theory

1. **Problem Statement:** (i) Let $G$ be a group and $g \in G$. Define the map $T_g : G \to G$ by $T_g(x) = xgx^{-1}$ for all $x \in G$. Show that $T_g$ is an automorphism of $G$.

1. **Problem Statement:** We consider the group of isometries $\mathrm{Isom}(\mathbb{R}^n)$ under composition.

1. **Problem statement:** We are given groups $A$ and $B$, a homomorphism $\theta : A \to \mathrm{Aut}(B)$, and a set $B \times_\theta A$ with operation defined by

1. The problem is understanding how to write cube and tetrahedron rotations as permutations of vertices or faces given an axis and an angle.\n\n2. For a cube with an axis through o