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| {{for|the finite simple group|Higman–Sims group}}
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| In mathematics, the '''Higman group''', introduced by {{harvs|txt|first=Graham|last=Higman|authorlink=Graham Higman|year=1951}}, was the first example of an infinite [[finitely presented group]] with no non-trivial finite quotients.
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| The quotient by the maximal proper [[normal subgroup]] is a [[Finitely generated group|finitely generated]] infinite [[simple group]]. {{harvtxt|Higman|1974}} later found some finitely presented infinite groups ''G''<sub>''n'',''r''</sub> that are simple if ''n'' is even and have a simple subgroup of index 2 if ''n'' is odd, one of which is one of the [[Thompson groups]].
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| Higman's group is generated by 4 elements ''a'', ''b'', ''c'', ''d'' with the relations
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| :<math>a^{-1}ba=b^2,\quad b^{-1}cb=c^2,\quad c^{-1}dc=d^2,\quad d^{-1}ad=a^2</math>. | |
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| ==References==
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| *{{Citation | last1=Higman | first1=Graham | author1-link=Graham Higman | title=A finitely generated infinite simple group | doi=10.1112/jlms/s1-26.1.59 | id={{MR|0038348}} | year=1951 | journal=Journal of the London Mathematical Society. Second Series | issn=0024-6107 | volume=26 | issue=1 | pages=61–64}}
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| *{{Citation | last1=Higman | first1=Graham | author1-link=Graham Higman | title=Finitely presented infinite simple groups | url=http://books.google.com/books?id=LPvuAAAAMAAJ | publisher=Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra | series=Notes on Pure Mathematics | isbn=978-0-7081-0300-5 | id={{MR|0376874}} | year=1974 | volume=8}}
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| [[Category:Group theory]]
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Revision as of 01:43, 27 February 2014
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