by $(A \otimes_{k} M_2(k))^G$. Moreover, can the construction can be generalised to any algebra or group? Our attempts to answer these questions form the basis of the work in this thesis.

### Twisting theory {#subsec: theoryoftwist}

Before stating some of our main results we must define the constructions involved in our work. Let us make the following assumptions that will remain in place until the end of §\[subsec: theoryoftwist\]. Our base field $k$ will be assumed to be algebraically closed and $A$ will denote an associative $k$-algebra with identity. The finite abelian group $G$ will satisfy $\text{char}(k) \nmid |G|$ and will act on $A$ by $k$-algebra automorphisms. We will also fix a duality isomorphism between $G$ and its group of characters $G^{\vee}$, where the character to which an element $g \in G$ is mapped is denoted by $\chi_g$.

#### Construction 1 {#subsubsec: constr1}

Let us define the first of two twisting constructions that we will use. There are two key components of Construction 1; first, the inducement of a $G$-grading on $A$ via the action of $G$, and second, the notion of using a normalised 2-cocycle to twist the $G$-graded multiplication in $A$.

The definition of a *$G$-graded algebra* is 