Group extension

This fact was a motivation for completing the classification of finite simple groups.

One extension, the direct product, is immediately obvious.

to be abelian groups, then the set of isomorphism classes of extensions of

is called the extension problem, and has been studied heavily since the late nineteenth century.

The classification of finite simple groups gives us a complete list of finite simple groups; so the solution to the extension problem would give us enough information to construct and classify all finite groups in general.

In general, this problem is very hard, and all the most useful results classify extensions that satisfy some additional condition.

We say that the extensions and are equivalent (or congruent) if there exists a group isomorphism

In fact it is sufficient to have a group homomorphism; due to the assumed commutativity of the diagram, the map

with quotient group isomorphic to the Klein four-group.

Split extensions are very easy to classify, because an extension is split if and only if the group G is a semidirect product of K and H. Semidirect products themselves are easy to classify, because they are in one-to-one correspondence with homomorphisms from

, where Aut(K) is the automorphism group of K. For a full discussion of why this is true, see semidirect product.

However, in group theory the opposite terminology has crept in, partly because of the notation

, which reads easily as extensions of Q by N, and the focus is on the group Q.

A paper of Ronald Brown and Timothy Porter on Otto Schreier's theory of nonabelian extensions uses the terminology that an extension of K gives a larger structure.

, the center of the group E. The set of isomorphism classes of central extensions of G by A is in one-to-one correspondence with the cohomology group

This kind of split example corresponds to the element 0 in

Similarly, the central extension of a Lie algebra

There is a general theory of central extensions in Maltsev varieties.

[4] There is a similar classification of all extensions of G by A in terms of homomorphisms from

, a tedious but explicitly checkable existence condition involving

[5] In Lie group theory, central extensions arise in connection with algebraic topology.

More precisely, a connected covering space G∗ of a connected Lie group G is naturally a central extension of G, in such a way that the projection is a group homomorphism, and surjective.

(The group structure on G∗ depends on the choice of an identity element mapping to the identity in G.) For example, when G∗ is the universal cover of G, the kernel of π is the fundamental group of G, which is known to be abelian (see H-space).

Conversely, given a Lie group G and a discrete central subgroup Z, the quotient G/Z is a Lie group and G is a covering space of it.

In the terminology of theoretical physics, generators of a are called central charges.

These generators are in the center of e; by Noether's theorem, generators of symmetry groups correspond to conserved quantities, referred to as charges.

The basic examples of central extensions as covering groups are: The case of SL2(R) involves a fundamental group that is infinite cyclic.

Here the central extension involved is well known in modular form theory, in the case of forms of weight ½.

A projective representation that corresponds is the Weil representation, constructed from the Fourier transform, in this case on the real line.

Metaplectic groups also occur in quantum mechanics.