Group Extension - Central Extension

A central extension of a group G is a short exact sequence of groups

such that A is in Z(E), the center of the group E. The set of isomorphism classes of central extensions of G by A (where G acts trivially on A) is in one-to-one correspondence with the cohomology group H2(G,A).

Examples of central extensions can be constructed by taking any group G and any abelian group A, and setting E to be A×G. This kind of split example (a split extension in the sense of the extension problem, since G is present as a subgroup of E) isn't of particular interest, since it corresponds to the element 0 in H2(G,A) under the above correspondence. More serious examples are found in the theory of projective representations, in cases where the projective representation cannot be lifted to an ordinary linear representation.

In the case of finite perfect groups, there is a universal perfect central extension.

Similarly, the central extension of a Lie algebra is an exact sequence

such that is in the center of .

There is a general theory of central extensions in Maltsev varieties, see the paper by Janelidze and Kelly listed below.