Editing Free group (section)

== History ==
Free groups first arose in the study of [[hyperbolic geometry]], as examples of [[Fuchsian group]]s (discrete groups acting by [[isometry|isometries]] on the [[Hyperbolic geometry|hyperbolic plane]]).  In an 1882 paper, [[Walther von Dyck]] pointed out that these groups have the simplest possible [[group presentation|presentations]].<ref>{{cite journal | last = von Dyck | first = Walther | author-link = Walther von Dyck | title = Gruppentheoretische Studien (Group-theoretical Studies) | journal = [[Mathematische Annalen]] | volume = 20 | issue = 1 | pages = 1–44 | year = 1882 | url = http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002246724&L=1 | doi = 10.1007/BF01443322 | s2cid = 179178038 | access-date = 2015-09-01 | archive-date = 2016-03-04 | archive-url = https://web.archive.org/web/20160304201754/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002246724&L=1 | url-status = dead }}</ref> The algebraic study of free groups was initiated by [[Jakob Nielsen (mathematician)|Jakob Nielsen]] in 1924, who gave them their name and established many of their basic properties.<ref>{{cite journal|last=Nielsen|first=Jakob|author-link=Jakob Nielsen (mathematician)|year=1917|title=Die Isomorphismen der allgemeinen unendlichen Gruppe mit zwei Erzeugenden|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002266873&L=1|journal=[[Mathematische Annalen]]|volume=78|issue=1|pages=385–397|doi=10.1007/BF01457113|jfm=46.0175.01|mr=1511907|s2cid=119726936|access-date=2015-09-01|archive-date=2016-03-05|archive-url=https://web.archive.org/web/20160305141749/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002266873&L=1|url-status=dead}}
<!-- note the journal volume was published in 1964, but its JFM review and the text of the article use the date 1917---><!-- note the journal volume was published in 1964, but its JFM review and the text of the article use the date 1917.--></ref><ref>{{cite journal | last = Nielsen | first = Jakob | author-link = Jakob Nielsen (mathematician) | title = On calculation with noncommutative factors and its application to group theory. (Translated from Danish) | journal = The Mathematical Scientist | volume = 6 (1981) | issue = 2 | pages = 73–85 | year = 1921 }}</ref><ref>{{cite journal | last = Nielsen | first = Jakob | author-link = Jakob Nielsen (mathematician) | title = Die Isomorphismengruppe der freien Gruppen | journal = Mathematische Annalen | volume = 91 | issue = 3 | pages = 169–209 | year = 1924 | url = http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002269813&L=1 | doi = 10.1007/BF01556078 | s2cid = 122577302 | access-date = 2015-09-01 | archive-date = 2016-03-05 | archive-url = https://web.archive.org/web/20160305073827/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002269813&L=1 | url-status = dead }}</ref> [[Max Dehn]] realized the connection with topology, and obtained the first proof of the full [[Nielsen–Schreier theorem]].<ref>See {{cite journal | last1 = Magnus | first1 = Wilhelm | author-link1 = Wilhelm Magnus | last2 = Moufang | first2 = Ruth | author-link2 = Ruth Moufang | title = Max Dehn zum Gedächtnis | journal = Mathematische Annalen | volume = 127 | issue = 1 | pages = 215–227 | year = 1954 | url = http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002283808&L=1 | doi = 10.1007/BF01361121 | s2cid = 119917209 | access-date = 2015-09-01 | archive-date = 2016-03-05 | archive-url = https://web.archive.org/web/20160305072926/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002283808&L=1 | url-status = dead }}</ref> [[Otto Schreier]] published an algebraic proof of this result in 1927,<ref>{{cite journal | last = Schreier | first = Otto | author-link = Otto Schreier | title = Die Untergruppen der freien Gruppen | journal = Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg | volume = 5 | year = 1928 | issue = 1 | pages = 161–183 | doi = 10.1007/BF02952517 | s2cid = 121888949 }}</ref> and [[Kurt Reidemeister]] included a comprehensive treatment of free groups in his 1932 book on [[combinatorial topology]].<ref>{{cite book | last = Reidemeister | first = Kurt | author-link = Kurt Reidemeister | title = Einführung in die kombinatorische Topologie | publisher = Wissenschaftliche Buchgesellschaft | date = 1972 |orig-date=1932 | location = Darmstadt}}</ref> Later on in the 1930s, [[Wilhelm Magnus]] discovered the connection between the [[lower central series]] of free groups and [[free Lie algebra]]s.