“Nottingham group”的意思、由来-开放百科全书

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Nottingham group

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In the mathematical field of infinite group theory, the Nottingham group is the group J(F_p) or N(F_p) consisting of formal power series t + a₂t²+...

with coefficients in F_p. The group multiplication is given by formal composition also called substitution. That is, if

and if is another element, then

.

The group multiplication is not abelian. The group was studied by number theorists as the group of wild automorphisms of the local field F_p((t)) and by group theorists including D. {{harvtxt|Johnson|1988}} and the name "Nottingham group" refers to his former domicile.

This group is a finitely generated pro-p-group, of finite width. For every finite group of order a power of p there is a closed subgroup of the Nottingham group isomorphic to that finite group.

References

{{Citation | last1=Johnson | first1=D. L. | title=The group of formal power series under substitution | url=http://anziamj.austms.org.au/JAMSA/V45/Part3/Johnson.html | mr=957195 | year=1988 | journal=Australian Mathematical Society. Journal. Series A. Pure Mathematics and Statistics | issn=0263-6115 | volume=45 | issue=3 | pages=296–302 | doi=10.1017/s1446788700031001}}
{{Citation | last1=Camina | first1=Rachel | editor1-last=du Sautoy | editor1-first=Marcus | editor2-last=Segal | editor2-first=Dan | editor3-last=Shalev | editor3-first=Aner | title=New horizons in pro-p groups | url=https://books.google.com/books?isbn=0817641718 | publisher=Birkhäuser Boston | location=Boston, MA | series=Progr. Math. | isbn=978-0-8176-4171-9 | mr=1765121 | year=2000 | volume=184 | chapter=The Nottingham group | pages=205–221}}
{{Citation | last1=Fesenko | first1=Ivan | title= On just infinite pro-p-groups and arithmetically profinite extensions | year=1999 | journal= J. fuer die reine und angew. Math. | volume=517 | pages=61-80}}
{{Citation | last1=du Sautoy| first1=Marcus | last2=Fesenko | first2=Ivan | editor1-last=du Sautoy | editor1-first=Marcus | editor2-last=Segal | editor2-first=Dan | editor3-last=Shalev | editor3-first=Aner | title=New horizons in pro-p groups | url=https://books.google.com/books?isbn=0817641718 | publisher=Birkhäuser Boston | location=Boston, MA | series=Progr. Math. | isbn=978-0-8176-4171-9 | mr=1765121 | year=2000 | volume=184 | chapter= Where the wild things are: ramification groups and the Nottingham group | pages=287–328}}

3 : Group theory|History of Nottingham|University of Nottingham

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