Group extension

In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence

Since any finite group G possesses a maximal normal subgroup N with simple factor group G/N, all finite groups may be constructed as a series of extensions with finite simple groups. This fact was a motivation for completing the classification of finite simple groups.

An extension is called a central extension if the subgroup N lies in the center of G.

One extension, the direct product, is immediately obvious. If one requires G and Q to be abelian groups, then the set of isomorphism classes of extensions of Q by a given (abelian) group N is in fact a group, which is isomorphic to

cf. the Ext functor. Several other general classes of extensions are known but no theory exists that treats all the possible extensions at one time. Group extension is usually described as a hard problem; it is termed the extension problem.

Solving the extension problem amounts to classifying all extensions of H by K; or more practically, by expressing all such extensions in terms of mathematical objects that are easier to understand and compute. In general, this problem is very hard, and all the most useful results classify extensions that satisfy some additional condition.

It is important to know when two extensions are equivalent or congruent. We say that the extensions

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.[3]

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

There is a general theory of central extensions in Maltsev varieties.[4]

In Lie group theory, central extensions arise in connection with algebraic topology. Roughly speaking, central extensions of Lie groups by discrete groups are the same as covering groups. 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.

More generally, when the groups A, E and G occurring in a central extension are Lie groups, and the maps between them are homomorphisms of Lie groups, then if the Lie algebra of G is g, that of A is a, and that of E is e, then e is a central Lie algebra extension of g by a. 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 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.