A group extension of by an abelian -module is a short exact sequencewhose conjugation action on is the specified -action.
Equivalence classes of group extensions of by an abelian -module correspond to . A section produces the extension cocycleand changing the section changes by a group coboundary.
An extension cocycle is the two-cocycle obtained from a section of a group extension. It measures the failure of the section to be a group homomorphism.
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Group extension is a concept in group theory, a branch of abstract algebra. It refers to the process of creating a new group from a known group by adding new elements that satisfy certain properties related to the original group. More formally, it describes a way to construct a group \( G \) that contains a normal subgroup \( N \) and a quotient group \( G/N \).