Editing Cayley's theorem (section)

== History ==
When Cayley (1854) introduced what are now called ''groups'', the modern definitions did not exist, and it was not immediately clear that this was equivalent to what were then called groups, which are now called ''permutation groups''. Cayley's theorem unifies the two.

Although Burnside<ref>{{Citation | last = Burnside | first = William | author-link = William Burnside | title = Theory of Groups of Finite Order | page = 22  | location = Cambridge | year = 1911 | edition = 2 | url = https://babel.hathitrust.org/cgi/pt?id=uc1.b4062919;view=1up;seq=52;size=125 | isbn = 0-486-49575-2}}</ref> 
attributes the theorem 
to [[Camille Jordan|Jordan]],<ref>{{Citation | last = Jordan | first = Camille | author-link = Camille Jordan | title = Traite des substitutions et des equations algebriques | publisher = Gauther-Villars | location = Paris | year = 1870}}</ref> 
Eric Nummela<ref>{{Citation | last = Nummela | first = Eric | title = Cayley's Theorem for Topological Groups | journal = American Mathematical Monthly | volume = 87 | issue = 3 | year = 1980 | pages = 202–203 | doi = 10.2307/2321608 | jstor = 2321608 | publisher = Mathematical Association of America}}</ref> 
nonetheless argues that the standard name&mdash;"Cayley's Theorem"&mdash;is in fact appropriate.  Cayley's original 1854 paper,<ref>{{Citation  | last = Cayley | first = Arthur | author-link = Arthur Cayley | title = On the theory of groups as depending on the symbolic equation θ<sup>n</sup>=1 | journal = Philosophical Magazine | volume = 7 | issue = 42 | pages = 40–47 | year = 1854 | url = https://books.google.com/books?id=_LYConosISUC&pg=PA40 }}</ref> 
showed that the correspondence in the theorem is one-to-one, but he did not explicitly show it was a homomorphism (and thus an embedding).  However, Nummela notes that Cayley made this result known to the mathematical community at the time, thus predating Jordan by 16 years or so.

The theorem was later published by [[Walther Dyck]] in 1882<ref>{{Citation | last=von Dyck | year=1882 | first=Walther | author-link=Walther Dyck | title=Gruppentheoretische Studien |trans-title=Group-theoretical Studies | url=https://archive.org/stream/mathematischean54behngoog#page/n38/mode/1up | doi=10.1007/BF01443322 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=20 | issue=1 | page=30| hdl=2027/njp.32101075301422 | s2cid=179178038 | hdl-access=free }}. {{in lang|de}}</ref> and is attributed to Dyck in the first edition of Burnside's book.<ref>{{Citation | last = Burnside | first = William | author-link = William Burnside | title = Theory of Groups of Finite Order | page = 22  | location = Cambridge | year = 1897 | edition = 1 | url = https://archive.org/stream/cu31924086163726#page/n43/mode/2up }}</ref>