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.
Read more about this topic: Group Extension
Famous quotes containing the words central and/or extension:
“Friends serve central functions for children that parents do not, and they play a critical role in shaping children’s social skills and their sense of identity. . . . The difference between a child with close friendships and a child who wants to make friends but is unable to can be the difference between a child who is happy and a child who is distressed in one large area of life.”
—Zick Rubin (20th century)
“Slavery is founded in the selfishness of man’s nature—opposition to it, is [in?] his love of justice.... Repeal the Missouri compromise—repeal all compromises—repeal the declaration of independence—repeal all past history, you still can not repeal human nature. It still will be the abundance of man’s heart, that slavery extension is wrong; and out of the abundance of his heart, his mouth will continue to speak.”
—Abraham Lincoln (1809–1865)