Group extension Information
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If and are two groups, then is an extension of by if there is a short exact sequence
If is an extension of by , then is a group, is a normal subgroup of and the quotient group is isomorphic to the group . Group extensions arise in the context of the extension problem, where the groups and are known and the properties of are to be determined. Note that the phrasing " is an extension of by " is also used by some.^{ [1]}
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 if the subgroup lies in the center of .
Extensions in general
One extension, the direct product, is immediately obvious. If one requires and to be abelian groups, then the set of isomorphism classes of extensions of by a given (abelian) group 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 , then is an extension of both and . More generally, if is a semidirect product of and , written as , then is an extension of by , so such products as the wreath product provide further examples of extensions.
Extension problem
The question of what groups are extensions of by 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 , where each is an extension of 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
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
and
are equivalent (or congruent) if there exists a group isomorphism making commutative the diagram of Figure 1. In fact it is sufficient to have a group homomorphism; due to the assumed commutativity of the diagram, the map is forced to be an isomorphism by the short five lemma.
Warning
It may happen that the extensions and are inequivalent but G and G' are isomorphic as groups. For instance, there are inequivalent extensions of the Klein four-group by ,^{ [2]} but there are, up to group isomorphism, only four groups of order containing a normal subgroup of order with quotient group isomorphic to the Klein four-group.
Trivial extensions
A trivial extension is an extension
that is equivalent to the extension
where the left and right arrows are respectively the inclusion and the projection of each factor of .
Classifying split extensions
A split extension is an extension
with a homomorphism such that going from H to G by s and then back to H by the quotient map of the short exact sequence induces the identity map on H i.e., . In this situation, it is usually said that s splits the above exact sequence.
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.
Warning on terminology
In general in mathematics, an extension of a structure K is usually regarded as a structure L of which K is a substructure. See for example field extension. 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.^{ [3]}
Central extension
A central extension of a group G is a short exact sequence of groups
such that A is included in , 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 .
Examples of central extensions can be constructed by taking any group G and any abelian group A, and setting E to be . This kind of split example corresponds to the element 0 in 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.^{ [4]}
Generalization to general extensions
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 and the cohomology group .^{ [5]}
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 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 basic examples of central extensions as covering groups are:
- the spin groups, which double cover the special orthogonal groups, which (in even dimension) doubly cover the projective orthogonal group.
- the metaplectic groups, which double cover the symplectic groups.
The case of SL_{2}(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
- Lie algebra extension
- Ring extension
- Virasoro algebra
- HNN extension
- Group contraction
- Extension of a topological group
References
- ^ group+extension#Definition in nLab Remark 2.2.
- ^ page no. 830, Dummit, David S., Foote, Richard M., Abstract algebra (Third edition), John Wiley & Sons, Inc., Hoboken, NJ (2004).
- ^ Brown, Ronald; Porter, Timothy (1996). "On the Schreier theory of non-abelian extensions: generalisations and computations". Proceedings of the Royal Irish Academy Sect A. 96 (2): 213–227. MR 1641218.
- ^ Janelidze, George; Kelly, Gregory Maxwell (2000). "Central extensions in Malt'sev varieties". Theory and Applications of Categories. 7 (10): 219–226. MR 1774075.
- ^ P. J. Morandi, Group Extensions and H^{3}. From his collection of short mathematical notes.
- Mac Lane, Saunders (1975), Homology, Classics in Mathematics, Springer Verlag, ISBN 3-540-58662-8
- R.L. Taylor, Covering groups of non connected topological groups, Proceedings of the American Mathematical Society, vol. 5 (1954), 753–768.
- R. Brown and O. Mucuk, Covering groups of non-connected topological groups revisited, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 115 (1994), 97–110.