wiki143 - User contributions [en]

Oxford Inter-Collegiate Christian Union

2024-10-16T12:08:49Z

192.76.8.67: /* OICCU President */ Made the display of these names consistent with that of all the others

{{Short description|Christian union}}
{{Use dmy dates|date=April 2022}}
{{primary sources|date=May 2017}}
The '''Oxford Inter-Collegiate Christian Union''', usually known as '''OICCU''' ({{IPAc-en|ˈ|ɔɪ|k|j|uː}} {{respell|OY|kew}}), is the world's second oldest university [[Christian Union (students)|Christian Union]] and is the [[University of Oxford]]'s most prominent student [[Christianity|Christian]] organisation. It was formed in 1879.

Due to the strength of the [[Oxford Movement]] and later the [[Oxford Group]]s (alternative Christian movements), [[evangelicalism|evangelical]] Christians in Oxford have generally faced a more pluriform environment than in [[University of Cambridge|Cambridge]], and OICCU has tended to follow the general lead of its Cambridge counterpart, the [[Cambridge Inter-Collegiate Christian Union]] (CICCU).

OICCU admits postgraduate students as well as undergraduates, although postgraduates are eligible only for associate membership, and their needs may be better served by the Oxford Graduate Christian Forum.

==Aims and purpose==

The OICCU vision is:
''Giving every student in Oxford University the chance to hear and respond to the Gospel of Jesus Christ'' <ref>{{cite web |url=https://www.oiccu.org |title = Oxford Inter-Collegiate Christian Union}}</ref>

The three aims of OICCU are:
* ''Presenting the claims of [[Jesus Christ]] to the University''
* ''Uniting in fellowship those who desire to witness for Christ, and to deepen their spiritual life''
* ''Promoting involvement in God's work worldwide''

===Relationship to the local church===

* The local church is a biblical principle which OICCU does not try to replace in the Christian's life: OICCU encourages all its members to also be part of a local church and to contribute to that family of believers.
* OICCU has the opportunity as a student run organisation to put on events uniquely geared to what its members think its fellow-students want.
* OICCU also has the opportunity to be prominent in the college communities in a way that churches cannot do.
* OICCU is inter-denominational, so its declaration of belief reflects what its members believe to be central to the gospel, and not secondary issues which Christians differ on.

===Activities and organisation===

* ''Personal Evangelism'' is key to what OICCU does - its members want to get alongside non-Christians and tell them about Christ.
* ''College Groups'' enable OICCU to put on events geared towards evangelising the individual college communities.<ref>{{cite web |url=https://www.oiccu.org/get-involved |title = Get Involved {{!}} OICCU}}</ref> There is a college Christian Union group in almost every undergraduate college in the University.<ref>{{Cite web |url=https://www.oiccu.org/find-your-college |title=Oiccu |access-date=8 May 2017 |archive-date=20 November 2016 |archive-url=https://web.archive.org/web/20161120171544/http://www.oiccu.org/find-your-college |url-status=dead }}</ref> ''Text-a-toastie'' is a popular college outreach event. The collegiate structure also enables students to have fellowship with a small group of believers, which helps in reaching out to the rest of the college. OICCU believes that when non-Christians see the Christian Union's members acting like a family and supporting one another it helps with evangelism.
* ''Central meetings'' mean that members of OICCU can be encouraged by seeing that there are many people who also want to witness for Christ in Oxford. OICCU has speakers, music and opportunities to hear about God's work in Oxford and further abroad.
* ''Monday Morning Prayer'': The members of OICCU pray together regularly,<ref>{{cite web |url=https://www.oiccu.org/get-involved |title = Get Involved {{!}} OICCU}}</ref> recognising their belief that everything they do is useless if God is not at work. Central events are a resource to back up personal evangelism; OICCU has weekly events like 'Friday Lunchtime Talks' and bigger events like the Carol Services.
* ''Events Week'': Since 1940, OICCU has held weeks of evangelistic events including [[Apologetics#Christianity|apologetics]] talks and a summary of the gospel. This now happens annually, with a larger series of evangelistic events organised every three years.
* ''Residentials'': OICCU hosts several annual residentials for its members, in particular the "Freshaway" event, which was first held in September 2016, and aims to help new students [[Freshman#England and Wales|("freshers")]] make Christian friends in the university, alongside seminars and talks to train and equip Christians for evangelism.<ref>https://www.uccf.org.uk/news/oiccu-freshaway-2016.htm {{Dead link|date=February 2022}}</ref>
* ''The Search'' is an event which aims to provide opportunities for open discussion between Christians and non-Christians on a variety of topics, and to allow Christians to share their beliefs with their friends. This event usually takes place in a local Oxford café.<ref>{{Cite web | url=http://oxfordstudent.com/2017/05/06/exploring-christianity-cornmarket-agnostic-insight-christian-union/ | title=Exploring Christianity on Cornmarket : An agnostic insight into the Christian Union | date=6 May 2017 }}</ref>

==Beliefs and affiliation==
OICCU adopts the doctrinal basis of [[UCCF]] (Universities and Colleges Christian Fellowship), an [[evangelicalism|evangelical]] Christian organisation with which OICCU is affiliated.<ref>{{Cite web | url=https://www.uccf.org.uk/find-your-cu/central/oxford-uni-inter-collegiate.htm | title=Oxford Uni Inter-Collegiate CU }}</ref> The doctrinal basis contains what evangelicals perceive as the [[Bible|biblical]] foundations of Christianity. UCCF is in turn affiliated with the [[International Fellowship of Evangelical Students]] (IFES).

==History==

===Foundation===
OICCU was modelled after the [[Cambridge Inter-Collegiate Christian Union]] (CICCU), founded two years earlier, but later incorporated a Daily Prayer Meeting established in [[Brasenose College, Oxford|Brasenose College]] in 1867. Like [[Wycliffe Hall]] (also 1877), it could be seen as a response to the University's abandonment of its previous officially [[Protestantism|Protestant]] position. The initial members included [[Francis Chavasse]], subsequently [[Bishop of Liverpool]] and founder of [[St Peter's College, Oxford|St Peter's College]].

===Relations with the SCM===
OICCU was a founder member of the [[World Student Christian Federation|Student Christian Movement]] and followed its lead in liberalizing its doctrine. In 1914 OICCU, along with the rest of the University, suspended its activities.

After World War I, the Oxford SCM was reestablished under that name, but those who held OICCU's original doctrinal position established a separate ''Oxford University Bible Union''. In 1925 the two agreed to merge, and the OUBU became the ''Devotional Union of the Student Christian Movement in Oxford''. However, the merger was not successful and in Michaelmas 1927, the Devotional Union committee voted to secede. The SCM gave them permission to use the old (1879) name and so OICCU was born anew, adopting the Doctrinal Basis of the new Inter-Varsity Fellowship of Evangelical Unions (now [[UCCF]]) in 1928.

During much of this period, OICCU used some of the buildings later incorporated into [[St Peter's College, Oxford|St Peter's College]]. However, after 1933 it had the use of the [[Northgate Hall]] (just opposite the [[Oxford Union]] on St Michael's Street).

===The Oxford Groups===
During the 1920s and 1930s, an American preacher named [[Frank N. D. Buchman]] drew a considerable following at Oxford. He emphasized the use of small groups (with Buchman-appointed leaders) where sins were publicly confessed and repented of. The movement taught that the [[Holy Spirit]] was to directly guide Christians. These small groups became known as [[Oxford Group]]s and later [[Moral Re-Armament]]. The emphasis on small groups and personal belief was inherited by [[Alcoholics Anonymous]].

Buchman was appealing directly to the OICCU constituency, and Julian Thornton-Duesbury (one of OICCU's supervising university teachers) became a noted Buchmanite. However, OICCU's student leadership distanced themselves from Buchman.

===1940s: Problems and Packer===
The [[International Fellowship of Evangelical Students]], the worldwide body to which OICCU belongs, was planned at a conference in Oxford in the late 1930s.

[[World War II]] forced those plans to be delayed. The greatly reduced number of students in Oxford obviously interfered with OICCU itself; one medical student had to serve as President for much more than the customary one year of office. However, the Union maintained daily prayer meetings (in termtime) throughout the War. Afterwards, a Standing Committee of prominent past members was established to ensure the Union's long-term continuity in such circumstances and in 1948 they became trustees of the Northgate Hall. The Standing Committee also has some reserve powers regarding the Doctrinal Basis, although they have never been used.

More positively, the prominent evangelical theologian [[J. I. Packer]] was converted to evangelical Christianity at an OICCU meeting in the 1940s, during his first week at the university. While a student member he was not regarded as doctrinally sound enough to join the Executive Committee. However, he was appointed Librarian, taking a particular interest in OICCU's selection of out-of-print [[Puritan]] books. In the following decade Packer, along with [[Martyn Lloyd-Jones]], led a revival of [[Puritan]] studies amongst British pastors. He returned to Oxford in 2004 as the guest of honour at the 125th Anniversary celebrations.

===Post-war era===
The 1950s saw OICCU at perhaps its greatest numerical strength, while the [[World Student Christian Federation|SCM]] was seen to have moved towards [[Marxism]].{{citation needed|date=February 2019}} One leading figure at this time was [[Michael Green (theologian)|Michael Green]] (President in 1952), who has been a leading evangelical in the [[Church of England]] and then the [[Anglican Communion]] since the 1960s. Green has taken a particular interest in promoting the [[Charismatic Movement]], including within OICCU.

In a slightly later generation, [[Tom Wright (theologian)|Tom Wright]] was the OICCU President (1970–71) and published his first book together with other members of his year's Executive Committee. The book was a plea for a conservative [[Calvinist]] doctrinal position, a position he has since modified.

A feature of the post-war years has been the custom of triennial missions which attempt to explain the gospel to every undergraduate. These missions can trace their history back to the visit of [[Dwight Moody]] and [[Ira D. Sankey]] in 1882, but the current model began with a 1940 mission led by [[Martyn Lloyd-Jones|Lloyd-Jones]]. Subsequent main speakers have included [[Michael Green (theologian)|Michael Green]], [[R. C. Lucas|Dick Lucas]] (long-time rector of [[St Helen's Bishopsgate]]), [[John Stott]]; one of Stott's series of talks was subsequently published as ''Basic Christianity'', and [[Tim Keller (pastor)|Tim Keller]].

OICCU membership has diminished since the middle part of the century, and now usually stands in the low hundreds — however formal membership is not needed to participate, and as of March 2006 OICCU's group membership on [[Facebook]] exceeded its official membership. The lease on the [[Northgate Hall]] was given up in the 1980s, and the Union has returned to the peripatetic existence of its earliest years, meeting in various church and public buildings around the city. Its archives are now held in the [[Bodleian Library]] and it has the use of a small store room at [[St Ebbe's, Oxford|St Ebbe's]] church and New Road Baptist Church.

==OICCU President==
{{columns-list|colwidth=15em|
*1879: George King
*1881: Frank Webster
*1882: William Talbot Rice
*1885: Vernon Bartlet
*1891: Thomas Ketchlee
*1892: Tom Alvarez
*1893: Edmund Elwin
*1894: Temple Gairdner
*1895: Willie Holland
*1897: Fergus McNeile
*1898: Robert Drury
*1905: Jack Woodhouse
*1907: Geoffrey Lunt
*1909: Nathaniel Micklem
*1919: Willoughby Habershon
*1920: Noel Palmer
*1921: Willoughby Habershon
*1922: Talbot Mohan
*1925: [[Verrier Elwin]]
*1926: Gordon Aldis
*1935: David Bentley-Taylor
*1938: Herbert Pope
*1943: David Mullins
*1947: [[Donald Wiseman]]
*1951: Michael Farrer
*1952: [[Michael Green (theologian)|Michael Green]]
*1953 Martin Peppiatt
*1955: Kenneth Habershon
*1956: J. D. Morris
*1957: [[Patrick Harris]]
*1958: H. W. J. Harland
*1959: David R. Catchpole
*1960: B. T. Lloyd
*1961: John G. Wesson
*1962: [[Graham Dow|G. Graham Dow]]
*1963: A. M. G. Dalzell
*1964: Ray A. E. Shilling
*1965: R. D. Toley
*1966: John Clarke
*1967: [[Denis Alexander]]
*1968: Chris M. N. Sugden
*1969: Richard Kennedy
*1970: [[N. T. Wright]]
*1971: Duncan Munro
*1972: Christopher Foster
*1973: Morey Thomas
*1974: [[Keith Sinclair (bishop)|Keith Sinclair]]
*1975: Lindsay Brown
*1977: Shaun Atkins
*1978: Iain Broomfield
*1979: Stephen Wright
*1980: Tim Saunders
*1981: Richard Dain
*1982: David Field
*1983: Hugh Skiel
*1984: David Gray
*1985: Mark May
*1986: Rick Simpson
*1987: Andy Buckler
*1988: Simon Cansdale
*1989: Steve Divall
*1990: Richard Frank
*1991: Nat Schluter
*1994: Stuart Cashman
*1995: James Ewins
*1996: Stephen Jones
*1997: Ed Reid
*1998: David Scoffield
*1999: Paul Murray
*1999: Martin Thornley
*2000: Tom Patrick
*2001: Joshua Hordern
*2002: Ed Boddam-Whetham
*2003: David 'Benny' Goodman
*2004: John Aldis
*2005: Edward Clark
*2006: Gregory Tarr
*2007: Daniel Tredget
*2008: David Meryon
*2009: Graham Thornton
*2010: Joel Harland
*2011: Robbie Strachan
*2012: Tim Bateman
*2013: Josh Peppiatt
*2014: Luke Cornelius
*2015: Pete Downing
*2016: Clem Faux
*2017: Sam Hodson
*2018: Abigail Marthinet-Payne
*2019: Emily Lobb
*2020: Stephen Lidbetter
*2021: Thomas Farlow
*2022: Nadia Ford
*2023: Mary Chen
*2024: Charis Patterson}}

*

== See also ==
{{Portal|Christianity|University of Oxford}}
* [[Holy Club]]
* [[Oxford University Newman Society]]

==References==
{{Reflist}}

==Bibliography==
* ''Born Anew'' John S. Reynolds : Oxford, OICCU Centenary & Executive Committees, 1979.
* ''Meeting Jesus at University: Rites of Passage and Student Evangelicals'' Edward Dutton: 2008. Ashgate.
* ''Christ and the Colleges'' F. [[Donald Coggan]] : London, [[Inter-Varsity Fellowship]], 1934.
* ''The Evangelicals at Oxford, 1735-1871 : a record of an unchronicled movement, with the record extended to 1905, and an essay on Oxford evangelical theology'' John S. Reynolds with [[J. I. Packer]] : Abingdon, [[Marcham Manor Press]], 1975.
* ''The Evangelicals at Oxford, 1735-1871 : a record of an unchronicled movement'' John S. Reynolds : Oxford, [[Blackwell's|Basil Blackwell]], 1953.
* ''From Cambridge to the world'': 125 years of student witness / [[Oliver R. Barclay]] and [[Robert M. Horn]] : Leicester, [[Inter-Varsity Press]], 2002, {{ISBN|0-85111-499-7}}.

==External links==
* [http://www.oiccu.org/ Oxford Inter-Collegiate Christian Union] website
* [http://graduatechristianunion.org/ Oxford University Graduate Christian Forum] website
* [http://www.e-n.org.uk/2537-It-began-with-prayer.htm A brief history of the OICCU] by [[Liam Beadle]], from the May 2004 edition of ''Evangelicals Now''

{{University of Oxford}}

[[Category:Religious organizations established in 1879]]
[[Category:Evangelicalism in the United Kingdom]]
[[Category:Clubs and societies of the University of Oxford|Christian Union, Inter-Collegiate]]
[[Category:Christian student societies in the United Kingdom]]

Commutator subgroup

2023-04-24T17:10:20Z

192.76.8.67: /* Examples */

{{short description|Smallest normal subgroup by which the quotient is commutative}}
In [[mathematics]], more specifically in [[abstract algebra]], the '''commutator subgroup''' or '''derived subgroup''' of a [[group (mathematics)|group]] is the [[subgroup (mathematics)|subgroup]] [[generating set of a group|generated]] by all the [[commutator]]s of the group.<ref>{{harvtxt|Dummit|Foote|2004}}</ref><ref>{{harvtxt|Lang|2002}}</ref>

The commutator subgroup is important because it is the [[Universal property|smallest]] [[normal subgroup]] such that the [[quotient group]] of the original group by this subgroup is [[abelian group|abelian]]. In other words, <math>G/N</math> is abelian [[if and only if]] <math>N</math> contains the commutator subgroup of <math>G</math>. So in some sense it provides a measure of how far the group is from being abelian; the larger the commutator subgroup is, the "less abelian" the group is.

== Commutators ==
{{main|Commutator}}
For elements <math>g</math> and <math>h</math> of a group ''G'', the [[commutator]] of <math>g</math> and <math>h</math> is <math>[g,h] = g^{-1}h^{-1}gh</math>. The commutator <math>[g,h]</math> is equal to the [[identity element]] ''e'' if and only if <math>gh = hg</math> , that is, if and only if <math>g</math> and <math>h</math> commute. In general, <math>gh = hg[g,h]</math>.

However, the notation is somewhat arbitrary and there is a non-equivalent variant definition for the commutator that has the inverses on the right hand side of the equation: <math>[g,h] = ghg^{-1}h^{-1}</math> in which case <math>gh \neq hg[g,h]</math> but instead <math>gh = [g,h]hg</math>.

An element of ''G'' of the form <math>[g,h]</math> for some ''g'' and ''h'' is called a commutator. The identity element ''e'' = [''e'',''e''] is always a commutator, and it is the only commutator if and only if ''G'' is abelian.

Here are some simple but useful commutator identities, true for any elements ''s'', ''g'', ''h'' of a group ''G'':

* <math>[g,h]^{-1} = [h,g],</math>
* <math>[g,h]^s = [g^s,h^s],</math> where <math>g^s = s^{-1}gs</math> (or, respectively, <math> g^s = sgs^{-1}</math>) is the [[Conjugacy class|conjugate]] of <math>g</math> by <math>s,</math>
* for any [[Group homomorphism|homomorphism]] <math>f: G \to H </math>, <math>f([g, h]) = [f(g), f(h)].</math>

The first and second identities imply that the [[Set (mathematics)|set]] of commutators in ''G'' is closed under inversion and conjugation. If in the third identity we take ''H'' = ''G'', we get that the set of commutators is stable under any [[endomorphism]] of ''G''. This is in fact a generalization of the second identity, since we can take ''f'' to be the conjugation [[automorphism]] on ''G'', <math> x \mapsto x^s </math>, to get the second identity.

However, the product of two or more commutators need not be a commutator. A generic example is [''a'',''b''][''c'',''d''] in the [[free group]] on ''a'',''b'',''c'',''d''. It is known that the least order of a finite group for which there exists two commutators whose product is not a commutator is 96; in fact there are two nonisomorphic groups of order 96 with this property.<ref>{{harvtxt|Suárez-Alvarez}}</ref>

== Definition ==
This motivates the definition of the '''commutator subgroup''' <math>[G, G]</math> (also called the '''derived subgroup''', and denoted <math>G'</math> or <math>G^{(1)}</math>) of ''G'': it is the subgroup [[generating set of a group|generated]] by all the commutators.

It follows from this definition that any element of <math>[G, G]</math> is of the form

:<math>[g_1,h_1] \cdots [g_n,h_n] </math>

for some [[natural number]] <math>n</math>, where the ''g''''i'' and ''h''''i'' are elements of ''G''. Moreover, since <math>([g_1,h_1] \cdots [g_n,h_n])^s = [g_1^s,h_1^s] \cdots [g_n^s,h_n^s]</math>, the commutator subgroup is normal in ''G''. For any homomorphism ''f'': ''G'' → ''H'',

:<math>f([g_1,h_1] \cdots [g_n,h_n]) = [f(g_1),f(h_1)] \cdots [f(g_n),f(h_n)]</math>,

so that <math>f([G,G]) \subseteq [H,H]</math>.

This shows that the commutator subgroup can be viewed as a [[functor]] on the [[category of groups]], some implications of which are explored below. Moreover, taking ''G'' = ''H'' it shows that the commutator subgroup is stable under every endomorphism of ''G'': that is, [''G'',''G''] is a [[fully characteristic subgroup]] of ''G'', a property considerably stronger than normality.

The commutator subgroup can also be defined as the set of elements ''g'' of the group that have an expression as a product ''g'' = ''g''1 ''g''2 ... ''g''''k'' that can be rearranged to give the identity.

=== Derived series ===
This construction can be iterated:
:<math>G^{(0)} := G</math>
:<math>G^{(n)} := [G^{(n-1)},G^{(n-1)}] \quad n \in \mathbf{N}</math>
The groups <math>G^{(2)}, G^{(3)}, \ldots</math> are called the '''second derived subgroup''', '''third derived subgroup''', and so forth, and the descending [[normal series]]
:<math>\cdots \triangleleft G^{(2)} \triangleleft G^{(1)} \triangleleft G^{(0)} = G</math>
is called the '''derived series'''. This should not be confused with the '''[[lower central series]]''', whose terms are <math>G_n := [G_{n-1},G]</math>.

For a finite group, the derived series terminates in a [[perfect group]], which may or may not be trivial. For an infinite group, the derived series need not terminate at a finite stage, and one can continue it to infinite [[ordinal number]]s via [[transfinite recursion]], thereby obtaining the '''transfinite derived series''', which eventually terminates at the [[perfect core]] of the group.

=== Abelianization ===
Given a group <math>G</math>, a [[quotient group]] <math>G/N</math> is abelian if and only if <math>[G, G]\subseteq N</math>.

The quotient <math>G/[G, G]</math> is an abelian group called the '''abelianization''' of <math>G</math> or <math>G</math> '''made abelian'''.<ref>{{harvtxt|Fraleigh|1976|p=108}}</ref> It is usually denoted by <math>G^{\operatorname{ab}}</math> or <math>G_{\operatorname{ab}}</math>.

There is a useful categorical interpretation of the map <math>\varphi: G \rightarrow G^{\operatorname{ab}}</math>. Namely <math>\varphi</math> is universal for homomorphisms from <math>G</math> to an abelian group <math>H</math>: for any abelian group <math>H</math> and homomorphism of groups <math>f: G \to H</math> there exists a unique homomorphism <math>F: G^{\operatorname{ab}}\to H</math> such that <math>f = F \circ \varphi</math>. As usual for objects defined by universal mapping properties, this shows the uniqueness of the abelianization <math>G^{\operatorname{ab}}</math> up to canonical isomorphism, whereas the explicit construction <math>G\to G/[G, G]</math> shows existence.

The abelianization functor is the [[adjoint functors|left adjoint]] of the inclusion functor from the [[category of abelian groups]] to the category of groups. The existence of the abelianization functor '''Grp''' → '''Ab''' makes the category '''Ab''' a [[reflective subcategory]] of the category of groups, defined as a full subcategory whose inclusion functor has a left adjoint.

Another important interpretation of <math>G^{\operatorname{ab}}</math> is as <math>H_1(G, \mathbb{Z})</math>, the first [[group homology|homology group]] of <math>G</math> with integral coefficients.

=== Classes of groups ===
A group <math>G</math> is an '''[[abelian group]]''' if and only if the derived group is trivial: [''G'',''G''] = {''e''}. Equivalently, if and only if the group equals its abelianization. See above for the definition of a group's abelianization.

A group <math>G</math> is a '''[[perfect group]]''' if and only if the derived group equals the group itself: [''G'',''G''] = ''G''. Equivalently, if and only if the abelianization of the group is trivial. This is "opposite" to abelian.

A group with <math>G^{(n)}=\{e\}</math> for some ''n'' in '''N''' is called a '''[[solvable group]]'''; this is weaker than abelian, which is the case ''n'' = 1.

A group with <math>G^{(n)} \neq \{e\}</math> for all ''n'' in '''N''' is called a '''non-solvable group'''.

A group with <math>G^{(\alpha)}=\{e\}</math> for some [[ordinal number]], possibly infinite, is called a '''[[perfect radical|hypoabelian group]]'''; this is weaker than solvable, which is the case ''α'' is finite (a natural number).

=== Perfect group ===
{{Main articles|Perfect group}}
Whenever a group <math>G</math> has derived subgroup equal to itself, <math>G^{(1)} =G</math>, it is called a '''perfect group'''. This includes non-abelian [[Simple group|simple groups]] and the [[Special linear group|special linear groups]] <math>\operatorname{SL}_n(k)</math> for a fixed field <math>k</math>.

== Examples ==
* The commutator subgroup of any [[abelian group]] is [[Trivial group|trivial]].
* The commutator subgroup of the [[general linear group]] <math>\operatorname{GL}_n(k)</math> over a [[Field (mathematics)|field]] or a [[division ring]] ''k'' equals the [[special linear group]] <math>\operatorname{SL}_n(k)</math> provided that <math>n \ne 2</math> or ''k'' is not the [[finite field|field with two elements]].<ref>{{citation|author=Suprunenko|first=D.A.|title=Matrix groups|publisher=American Mathematical Society|year=1976|series=Translations of Mathematical Monographs}}, Theorem II.9.4</ref>
* The commutator subgroup of the [[alternating group]] ''A''4 is the [[Klein four group]].
* The commutator subgroup of the [[symmetric group]] ''Sn'' is the [[alternating group]] ''An''.
* The commutator subgroup of the [[quaternion group]] ''Q'' = {1, −1, ''i'', −''i'', ''j'', −''j'', ''k'', −''k''} is [''Q'',''Q''] = {1, −1}.

=== Map from Out ===
Since the derived subgroup is [[Characteristic subgroup|characteristic]], any automorphism of ''G'' induces an automorphism of the abelianization. Since the abelianization is abelian, [[inner automorphism]]s act trivially, hence this yields a map
:<math>\operatorname{Out}(G) \to \operatorname{Aut}(G^{\mbox{ab}})</math>

==See also==
*[[Solvable group]]
*[[Nilpotent group]]
*The abelianization ''H''/''H''<nowiki>'</nowiki> of a subgroup ''H'' < ''G'' of finite [[Index of a subgroup|index]] (''G'':''H'') is the [[Artin transfer (group theory)#Artin transfer|target of the Artin transfer]] ''T''(''G'',''H'').

== Notes ==
<references/>

== References ==
* {{ citation | last1 = Dummit | first1 = David S. | last2 = Foote | first2 = Richard M. | title = Abstract Algebra | publisher = [[John Wiley & Sons]] | year = 2004 | edition = 3rd | isbn = 0-471-43334-9 }}
* {{ citation | first1 = John B. | last1 = Fraleigh | year = 1976 | isbn = 0-201-01984-1 | title = A First Course In Abstract Algebra | edition = 2nd | publisher = [[Addison-Wesley]] | location = Reading }}
* {{citation | last = Lang | first = Serge | author-link = Serge Lang | title = Algebra | publisher = [[Springer Science+Business Media|Springer]] | series = [[Graduate Texts in Mathematics]] | year = 2002 | isbn = 0-387-95385-X}}
* {{ cite web | url = https://math.stackexchange.com/q/7811 | first = Mariano | last = Suárez-Alvarez | title = Derived Subgroups and Commutators }}

==External links==
* {{springer|title=Commutator subgroup|id=p/c023440}}

[[Category:Group theory]]
[[Category:Functional subgroups]]
[[Category:Articles containing proofs]]
[[Category:Subgroup properties]]