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Wikipedia / 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 is an extension if there is a short exact sequence

is a normal subgroup and the quotient group is isomorphic. Group extensions arise in the context of the extension problem" is also used by some.

Since any finite group possesses a maximal normal subgroup with simple factor group, 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 lies in the center.

Extensions in general

One extension, the direct product to be abelian groups 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.

To consider some examples, if is a semidirect product, so such products as the wreath product provide further examples of extensions.

Extension problem

is called the extension problem, and has been studied heavily since the late nineteenth century. As to its motivation, consider that the composition series by some simple group. 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.

Classifying extensions

; 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

and

are equivalent is forced to be an isomorphism by the short five lemma.

Warning

inequivalent extensions of the Klein four-group, with quotient group isomorphic to the Klein four-group.

Trivial extensions

A trivial extension is an extension

that is equivalent to the extension

.

Classifying split extensions

A split extension is an extension

with a homomorphism by the quotient map of the short exact sequence induces the identity map splits the above exact sequence.

Split extensions are very easy to classify, because an extension is split if and only if is a semidirect product is the automorphism. For a full discussion of why this is true, see semidirect product.

Warning on terminology

is a substructure. See for example field extension.

A paper of Ronald Brown and Timothy Porter on Otto Schreier gives a larger structure.

Central extension

A central extension is a short exact sequence of groups

, the center is in one-to-one correspondence with the cohomology.

and any abelian group. This kind of split 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

.

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

Generalization to general extensions

, a tedious but explicitly checkable existence condition involving and the cohomology group .

Lie groups

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 of a connected Lie group is naturally a central extension of , in such a way that the projection

is a group homomorphism, and surjective. (The group structure on depends on the choice of an identity element mapping to the identity in .) For example, when is the universal cover of , the kernel of π is the fundamental group of , which is known to be abelian (see H-space). Conversely, given a Lie group and a discrete central subgroup , the quotient is a Lie group and is a covering space of it.

More generally, when the groups , and occurring in a central extension are Lie groups, and the maps between them are homomorphisms of Lie groups, then if the Lie algebra of is , that of is , and that of is , then is a central Lie algebra extension of by . In the terminology of theoretical physics, generators of are called central charges. These generators are in the center of ; 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 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 . 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

References

Further reading