In mathematics, a Gassmann triple (or Gassmann-Sunada triple) is a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
''G'' together with two
faithful actions on
sets ''X'' and ''Y'', such that ''X'' and ''Y'' are not
isomorphic as ''G''-sets but every element of ''G'' has the same number of
fixed points on ''X'' and ''Y''. They were introduced by
Fritz Gassmann
Fritz Gassmann (1899–1990) was a Swiss mathematician and geophysicist.
Life
His Ph.D. advisors at ETH Zurich were George Pólya and Hermann Weyl. He was a geophysics professor at the ETH Zurich.
Legacy
Gassmann is the eponym for the Gas ...
in 1926.
Applications
Gassmann triples have been used to construct examples of pairs of mathematical objects with the same invariants that are not isomorphic, including
arithmetically equivalent number fields and
isospectral graphs and
isospectral Riemannian manifolds.
Examples
The
simple group
SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service.
The d ...
''G'' =
SL3(F2) of order 168 acts on the
projective plane of order 2, and the actions on the 7 points and 7 lines give a Gassmann triple.
References
*
*
* {{Citation , authorlink=Toshikazu Sunada , doi=10.2307/1971195 , first=T., last= Sunada , title=Riemannian coverings and isospectral manifolds , journal=Annals of Mathematics , volume=121 , issue=1 , year=1985 , pages=169–186 , postscript= , jstor=1971195
Permutation groups