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 <math>Q</math> and <math>N</math> are two groups, then <math>G</math> is an extension of <math>Q</math> by <math>N</math> if there is a short exact sequence
- <math>1\to N\;\overset{\iota}{\to}\;G\;\overset{\pi}{\to}\;Q \to 1.</math>
If <math>G</math> is an extension of <math>Q</math> by <math>N</math>, then <math>G</math> is a group, <math>\iota(N)</math> is a normal subgroup of <math>G</math> and the quotient group <math>G/\iota(N)</math> is isomorphic to the group <math>Q</math>. Group extensions arise in the context of the extension problem, where the groups <math>Q</math> and <math>N</math> are known and the properties of <math>G</math> are to be determined. Note that the phrasing "<math>G</math> is an extension of <math>N</math> by <math>Q</math>" is also used by some.<ref>Template:Nlab Remark 2.2.</ref>
Since any finite group <math>G</math> possesses a maximal normal subgroup <math>N</math> with simple factor group <math>G/\iota(N)</math>, 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 <math>N</math> lies in the center of <math>G</math>.
Extensions in generalEdit
One extension, the direct product, is immediately obvious. If one requires <math>G</math> and <math>Q</math> to be abelian groups, then the set of isomorphism classes of extensions of <math>Q</math> by a given (abelian) group <math>N</math> is in fact a group, which is isomorphic to
- <math>\operatorname{Ext}^1_{\mathbb Z}(Q,N);</math>
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 Template:Nowrap, then <math>G</math> is an extension of both <math>H</math> and <math>K</math>. More generally, if <math>G</math> is a semidirect product of <math>K</math> and <math>H</math>, written as <math>G=K\rtimes H</math>, then <math>G</math> is an extension of <math>H</math> by <math>K</math>, so such products as the wreath product provide further examples of extensions.
Extension problemEdit
The question of what groups <math>G</math> are extensions of <math>H</math> by <math>N</math> 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 <math>\{A_i\}</math>, where each <math>\{A_{i+1}\}</math> is an extension of <math>\{A_i\}</math> 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 extensionsEdit
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
- <math>1 \to K\stackrel{i}{{}\to{}} G\stackrel{\pi}{{}\to{}} H\to 1</math>
and
- <math>1\to K\stackrel{i'}{{}\to{}} G'\stackrel{\pi'}{{}\to{}} H\to 1</math>
are equivalent (or congruent) if there exists a group isomorphism <math>T: G\to G'</math> making commutative the diagram below. In fact it is sufficient to have a group homomorphism; due to the assumed commutativity of the diagram, the map <math>T</math> is forced to be an isomorphism by the short five lemma.
WarningEdit
It may happen that the extensions <math>1\to K\to G\to H\to 1</math> and <math>1\to K\to G^\prime\to H\to 1</math> are inequivalent but G and G' are isomorphic as groups. For instance, there are <math>8</math> inequivalent extensions of the Klein four-group by <math>\mathbb{Z}/2\mathbb{Z}</math>,<ref>page no. 830, Dummit, David S., Foote, Richard M., Abstract algebra (Third edition), John Wiley & Sons, Inc., Hoboken, NJ (2004).</ref> but there are, up to group isomorphism, only four groups of order <math>8</math> containing a normal subgroup of order <math>2</math> with quotient group isomorphic to the Klein four-group.
Trivial extensionsEdit
A trivial extension is an extension
- <math>1\to K\to G\to H\to 1</math>
that is equivalent to the extension
- <math>1\to K\to K\times H\to H\to 1</math>
where the left and right arrows are respectively the inclusion and the projection of each factor of <math>K\times H</math>.
Classifying split extensionsEdit
A split extension is an extension
- <math>1\to K\to G\to H\to 1</math>
with a homomorphism <math>s\colon H \to G</math> 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., <math>\pi \circ s=\mathrm{id}_H</math>. 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 <math>H\to\operatorname{Aut}(K)</math>, where Aut(K) is the automorphism group of K. For a full discussion of why this is true, see semidirect product.
Warning on terminologyEdit
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 <math>\operatorname{Ext}(Q,N)</math>, 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.<ref>Template:Cite journal</ref>
Central extensionEdit
A central extension of a group G is a short exact sequence of groups
- <math>1\to A\to E\to G\to 1</math>
such that A is included in <math>Z(E)</math>, 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 <math>H^2(G,A)</math>.
Examples of central extensions can be constructed by taking any group G and any abelian group A, and setting E to be <math>A\times G</math>. This kind of split example corresponds to the element 0 in <math>H^2(G,A)</math> 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 <math>\mathfrak{g}</math> is an exact sequence
- <math>0\rightarrow \mathfrak{a}\rightarrow\mathfrak{e}\rightarrow\mathfrak{g}\rightarrow 0</math>
such that <math>\mathfrak{a}</math> is in the center of <math>\mathfrak{e}</math>.
There is a general theory of central extensions in Maltsev varieties.<ref>Template:Cite journal</ref>
Generalization to general extensionsEdit
There is a similar classification of all extensions of G by A in terms of homomorphisms from <math>G\to\operatorname{Out}(A)</math>, a tedious but explicitly checkable existence condition involving Template:Nowrap and the cohomology group Template:Nowrap.<ref>P. J. Morandi, Group Extensions and H3 Template:Webarchive. From his collection of short mathematical notes.</ref>
Lie groupsEdit
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 Template:Math of a connected Lie group Template:Math is naturally a central extension of Template:Math, in such a way that the projection
- <math>\pi\colon G^* \to G</math>
is a group homomorphism, and surjective. (The group structure on Template:Math depends on the choice of an identity element mapping to the identity in Template:Math.) For example, when Template:Math is the universal cover of Template:Math, the kernel of π is the fundamental group of Template:Math, which is known to be abelian (see H-space). Conversely, given a Lie group Template:Math and a discrete central subgroup Template:Math, the quotient Template:Math is a Lie group and Template:Math is a covering space of it.
More generally, when the groups Template:Math, Template:Math and Template:Math occurring in a central extension are Lie groups, and the maps between them are homomorphisms of Lie groups, then if the Lie algebra of Template:Math is Template:Math, that of Template:Math is Template:Math, and that of Template:Math is Template:Math, then Template:Math is a central Lie algebra extension of Template:Math by Template:Math. In the terminology of theoretical physics, generators of Template:Math are called central charges. These generators are in the center of Template:Math; 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 Template:Math 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 Template:Math. 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 alsoEdit
- Algebra extension
- Lie algebra extension
- Virasoro algebra
- HNN extension
- Group contraction
- Extension of a topological group