$G$, that is $\text{char}(k) \nmid |G|$. A normalised 2-cocycle will be denoted by $\mu$. Unadorned tensor products $\otimes$ will mean tensor products over $k$, $\otimes_k$.

### Notation {#subsec: notation}

In this section we will describe our basic notation that will be in place throughout the rest of the thesis, unless otherwise stated. More specialised notation may subsequently be defined as and when it is needed.

#### Group actions

The group of *algebra automorphisms* of $A$ is denoted by $\text{Aut}_{\text{alg}}(A)$, while the group of *group automorphisms* of $G$ is denoted by $\text{Aut}_{\text{grp}}(G)$. One of our base assumptions in §\[subsec: theoryoftwist\] was that $G$ acts on $A$ by $k$-algebra automorphisms, i.e. $G
\rightarrow \text{Aut}_{\text{alg}}(A)$. As $G$ is abelian we can assume that the group acts on the right without loss of generality. We will denote the action of an element $g \in G$ by $-^g$. The embedding of $G$ in the automorphism group means that for all $a,b \in A$, $g \in G$ and $\alpha \in k$ one has $(\alpha a)^g=\alpha a^g$ and $(ab)^g=a^gb^g$.

We will also define actions of $G$ on objects other than algebras later in the thesis. The same superscript notation will be used in such circumstances, although the 