Editing Group extension (section)

==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>