Editing Group extension

{{Short description|Group for which a given group is a normal subgroup}}
[[File:Group extension illustration.svg|thumb|400x400px|Extension of <math>Q</math> by <math>N</math>, resulting in the group <math>G</math>. They form a short exact sequence <math>1\to N\;\overset{\iota}{\to}\;G\;\overset{\pi}{\to}\;Q \to 1</math>. The [[Injective function|injective]] homomorphism <math>\iota</math> maps <math>N</math> to a normal subgroup of <math>G</math>. In turn, <math>\pi</math> maps <math>G</math> [[Surjective function|onto]] <math>Q</math>, sending each [[coset]] of <math>\iota(N)</math> to a different element of <math>Q</math>.]]
In [[mathematics]], a '''group extension''' is a general means of describing a [[group (mathematics)|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>{{nlab|id=group+extension#Definition}} Remark 2.2.</ref>

Since any [[finite group]] <math>G</math> possesses a [[Maximal subgroup|maximal]] [[normal subgroup]] <math>N</math> with [[simple group|simple]] [[factor group]] <math>G/\iota(N)</math>, all finite groups may be constructed as a series of extensions with finite [[simple group]]s. 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 a group|center]] of <math>G</math>.

==Extensions in general==

One extension, the [[direct product of groups|direct product]], is immediately obvious.  If one requires <math>G</math> and <math>Q</math> to be [[abelian group]]s, 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 {{nowrap|<math>G=K\times H</math>}}, 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 problem===

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 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  
:<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]].
[[File:Equivalence_of_group_extensions.svg|center|397x397px]]

====Warning====

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 extensions====
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 extensions====

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 function|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 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 <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 (mathematician)|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>{{cite journal|first1=Ronald|author-link1=Ronald Brown (mathematician)|last1= Brown |first2=Timothy|last2= Porter|title=On the Schreier theory of non-abelian extensions: generalisations and computations| journal=Proceedings of the Royal Irish Academy Sect A|volume=96 |year=1996|issue=2|pages=213–227|mr=1641218}}</ref>

==Central extension==
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 a group|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 [[Group cohomology|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 exact sequence|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 representation]]s, in cases where the projective representation cannot be lifted to an ordinary [[linear representation]].

In the case of finite [[perfect group]]s, 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 [[Quotient (universal algebra)|Maltsev varieties]].<ref>{{cite journal|first1=George|last1=Janelidze |first2= Gregory Maxwell|last2= Kelly|author2-link=Max Kelly|url=http://www.tac.mta.ca/tac/volumes/7/n10/7-10abs.html|title= Central extensions in Malt'sev varieties|journal=Theory and Applications of Categories|volume=7|year=2000|issue=10|pages=219–226|mr=1774075}}</ref>

===Generalization to general extensions===

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 {{nowrap|<math>H^3(G, Z(A))</math>}} and the cohomology group {{nowrap|<math>H^2(G, Z(A))</math>}}.<ref>P. J. Morandi, [http://sierra.nmsu.edu/morandi/notes/GroupExtensions.pdf Group Extensions and ''H''<sup>3</sup>] {{Webarchive|url=https://web.archive.org/web/20180517072932/http://sierra.nmsu.edu/morandi/notes/GroupExtensions.pdf |date=2018-05-17 }}. From his collection of short mathematical notes.</ref>

===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 group]]s. More precisely, a [[connected space|connected]] [[covering space]] {{math|''G''<sup>∗</sup>}} of a connected Lie group {{math|''G''}} is naturally a central extension of {{math|''G''}}, in such a way that the projection

:<math>\pi\colon G^* \to G</math>

is a group homomorphism, and surjective. (The group structure on {{math|''G''<sup>∗</sup>}} depends on the choice of an identity element mapping to the identity in {{math|''G''}}.) For example, when {{math|''G''<sup>∗</sup>}} is the [[universal cover]] of {{math|''G''}}, the kernel of ''π'' is the [[fundamental group]] of {{math|''G''}}, which is known to be abelian (see [[H-space]]). Conversely, given a Lie group {{math|''G''}} and a discrete central subgroup {{math|''Z''}}, the quotient {{math|''G''/''Z''}} is a Lie group and {{math|''G''}} is a covering space of it.

More generally, when the groups {{math|''A''}}, {{math|''E''}} and {{math|''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 {{math|''G''}} is {{math|'''g'''}}, that of {{math|''A''}} is {{math|'''a'''}}, and that of {{math|''E''}} is {{math|'''e'''}}, then {{math|'''e'''}} is a [[Lie algebra extension#Central extension|central Lie algebra extension]] of {{math|'''g'''}} by {{math|'''a'''}}.   In the terminology of [[theoretical physics]], generators of {{math|'''a'''}} are called [[central charge]]s.  These generators are in the center of {{math|'''e'''}}; by [[Noether's theorem]], generators of symmetry groups correspond to conserved quantities, referred to as [[charge (physics)|charges]].

The basic examples of central extensions as covering groups are:
* the [[spin group]]s, which double cover the [[special orthogonal group]]s, which (in even dimension) doubly cover the [[projective orthogonal group]].
* the [[metaplectic group]]s, which double cover the [[symplectic group]]s.
The case of {{math|[[SL2(R)|SL<sub>2</sub>('''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 {{math|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==
*[[Algebra extension]]
*[[Lie algebra extension]]
*[[Virasoro algebra]]
*[[HNN extension]]
*[[Group contraction]]
*[[Extension of a topological group]]

==References==
{{reflist}}

==Further reading==
*{{citation | first1=Saunders | last1=Mac Lane |authorlink = Saunders Mac Lane| title=Homology | publisher=[[Springer Verlag]] | year=1975 | isbn=3-540-58662-8 | series=Classics in Mathematics}}
*{{citation |last1=Taylor | first1=R. L.
| title=Covering  groups  of  non  connected topological  groups
| journal=[[Proceedings of the American Mathematical Society]]
| volume=5
| issue=5
| date=1954
| pages=753–768
| doi=10.1090/S0002-9939-1954-0087028-0 | doi-access=free}}
*{{citation
| last1=Brown | first1=R.
| last2=Mucuk | first2=O.
| title=Covering groups of non-connected topological groups revisited
| journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]]
| volume=115
| issue=1
| date=1994
| pages=97–110
| doi=10.1017/S0305004100071942}}

[[Category:Group theory]]