students I have known at Manchester, particularly those with whom I have shared an office. Their willingness to discuss maths, other ‘enlightening’ conversations and numerous distractions have all made my time as a PhD student thoroughly enjoyable.

Finally, I would like to thank my family and my girlfriend Kirsti – jag älskar dig. I’m very lucky to have you, and I preemptively forgive you if you give up reading after a couple of pages/paragraphs/words…

Introduction {#chap: introduction}
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Overview {#sec: introoverview}
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In this thesis we study a twisting operation on algebras. This operation is closely related to the classical notion of using a 2-cocycle to deform the multiplication in a group-graded algebra. Our primary concern is to establish whether certain properties are preserved under the twisting operation, the first results in this direction being proved in [@zhang1998twisted] and [@montgomery2005algebra]. In the former the twists we study are phrased as Zhang twists, while Montgomery’s results in the latter hold for generalised twists in the setting of Hopf algebras.

We use another construction of these *cocycle twists* that was independently described in [@bazlov2012cocycle]. This construction gives one much greater traction when trying to prove that properties are preserved under twisting, since 