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

1rightarrow Nrightarrow Grightarrow Qrightarrow 1. ,!

If G is an extension of Q by N, then G is a group, N is a normal subgroup of G and the quotient group G/N is isomorphic to group Q. Group extensions arise in the context of the extension problem, where the groups Q and N are known and the properties of G are to be determined.

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

Extensions in general

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 operatorname{Ext}^1_{mathbb Z}(Q,N); cf. the Ext functor. Several other general classes of extensions are known but no theory exists which treats all the possible extensions at one time. Group extension is usually described as a hard problem; it is termed the extension problem.

To consider some examples, if G = H × K, then G is an extension of both H and K. More generally, if G is a semidirect product of K and H, then G is an extension of H by K, so such products as the wreath product provide further examples of extensions.

Extension problem

The question of what groups G are extensions of H is called the extension problem, and has been studied heavily since the late nineteenth century. As to its motivation, consider that the composition series of a finite group is a finite sequence of subgroups {Ai}, where each Ai+1 is an extension of Ai by some simple group. The classification of finite simple groups would give 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.

We can use the language of diagrams to provide a more flexible definition of extension: a group G is an extension of a group H by a group K if and only if there is an exact sequence:

1rightarrow Krightarrow Grightarrow Hrightarrow 1

where 1 denotes the trivial group with a single element. This definition is more general in that it does not require that K be a subgroup of G; instead, K is isomorphic to a normal subgroup K* of G, and H is isomorphic to G/K*.

Classifying extensions

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.

Classifying split extensions

A split extension is an extension

1rightarrow Krightarrow Grightarrow Hrightarrow 1

such that there is a homomorphism scolon H rightarrow G such that going from H to G by s and then back to H by the quotient map induces the identity map on H. In this situation, it is usually said that s splits the above exact sequence.

Split extensions are very easy to classify, because the splitting lemma states that 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 Htooperatorname{Aut}(K), where Aut(K) is the automorphism group of K. For a full discussion of why this is true, see semidirect product.

Central extension

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

1rightarrow Arightarrow Erightarrow Grightarrow 1
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 is in one-to-one correspondence with the cohomology group H2(G,A), where the action of G on A is trivial.

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. 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

0rightarrow mathfrak{a}rightarrowmathfrak{e}rightarrowmathfrak{g}rightarrow 0
such that mathfrak{a} is in the center of mathfrak{e}.

Lie groups

Central extensions of Lie groups are identical to covering spaces of Lie groups.

If the group G is a Lie group, then a central extension of G is a Lie group as well, and the Lie algebra of a central extension of G is a central extension of the Lie algebra of G. In the terminology of theoretical physics, the generators of E not included in G are called central charges. These generators are in the center of the Lie algebra of E; by Noether's theorem, generators of symmetry groups correspond to conserved quantities, referred to as charges.

In Lie group theory, central extensions arise in connection with algebraic topology. Suppose G is a connected Lie group that is not simply connected. Its universal cover G* is again a Lie group, in such a way that the projection

π: G* → G

is a group homomorphism, and surjective. Its kernel is (up to isomorphism) the fundamental group of G; this is known to be abelian (see H-space). This construction gives rise to central extensions.

Conversely, given a Lie group G, with non-trivial center Z, the quotient G/Z is a Lie group and G is a central extension of it.

The basic examples 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.

See also


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