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:
“It’s easy to forget how central the French people are in everything we mean when we say Europe.”
—John Dos Passos (1896–1970)
“We are now a nation of people in daily contact with strangers. Thanks to mass transportation, school administrators and teachers often live many miles from the neighborhood schoolhouse. They are no longer in daily informal contact with parents, ministers, and other institution leaders . . . [and are] no longer a natural extension of parental authority.”
—James P. Comer (20th century)