An application of Burnside rings in elementary finite group theory.
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Dress, Andreas W.M., Siebeneicher, Christian, Yoshida, Tomoyuki (1990)
Séminaire Lotharingien de Combinatoire [electronic only]
Lewis, David W. (2001)
International Journal of Mathematics and Mathematical Sciences
Liufeng Cao, Dong Su, Hua Yao (2023)
Czechoslovak Mathematical Journal
Let be the Green ring of the weak Hopf algebra corresponding to Sweedler’s 4-dimensional Hopf algebra , and let be the automorphism group of the Green algebra . We show that the quotient group , where contains the identity map and is isomorphic to the infinite group and is the symmetric group of order 6.
Dong Su, Shilin Yang (2018)
Czechoslovak Mathematical Journal
Let be the unique noncommutative and noncocommutative eight dimensional semi-simple Hopf algebra. We first construct a weak Hopf algebra based on , then we investigate the structure of the representation ring of . Finally, we prove that the automorphism group of is just isomorphic to , where is the dihedral group with order 12.
Alex Bartel, Tim Dokchitser (2015)
Journal of the European Mathematical Society
If is a non-cyclic finite group, non-isomorphic -sets may give rise to isomorphic permutation representations . Equivalently, the map from the Burnside ring to the rational representation ring of has a kernel. Its elements are called Brauer relations, and the purpose of this paper is to classify them in all finite groups, extending the Tornehave–Bouc classification in the case of -groups.
Kenneth K. Nwabueze (1994)
Acta Mathematica et Informatica Universitatis Ostraviensis
Lawson, Tyler (2006)
Algebraic & Geometric Topology
Jon F. Carlson, Nadia Mazza, Jacques Thévenaz (2013)
Journal of the European Mathematical Society
Let be a finite group with a Sylow 2-subgroup which is either quaternion or semi-dihedral. Let be an algebraically closed field of characteristic 2. We prove the existence of exotic endotrivial -modules, whose restrictions to are isomorphic to the direct sum of the known exotic endotrivial -modules and some projective modules. This provides a description of the group of endotrivial -modules.
Kuku, Aderemi (2006)
Beiträge zur Algebra und Geometrie
Bouc, Serge (2002)
AMA. Algebra Montpellier Announcements [electronic only]
Jingcheng Dong (2010)
Czechoslovak Mathematical Journal
Let be a group algebra, and its quantum double. We first prove that the structure of the Grothendieck ring of can be induced from the Grothendieck ring of centralizers of representatives of conjugate classes of . As a special case, we then give an application to the group algebra , where is a field of characteristic and is a dihedral group of order .
Richard Dipper, Jie Du (1993)
Journal für die reine und angewandte Mathematik
Mikhail Khovanov (2014)
Fundamenta Mathematicae
A new calculus of planar diagrams involving diagrammatics for biadjoint functors and degenerate affine Hecke algebras is introduced. The calculus leads to an additive monoidal category whose Grothendieck ring contains an integral form of the Heisenberg algebra in infinitely many variables. We construct bases of the vector spaces of morphisms between products of generating objects in this category.
Stephen Donkin (1992)
Inventiones mathematicae
Panchadcharam, E., Street, R. (2007)
Journal of Homotopy and Related Structures
Paul Balmer (2013)
Journal of the European Mathematical Society
Let be a field of characteristic . Let be a finite group of order divisible by and a -Sylow subgroup of . We describe the kernel of the restriction homomorphism , for the group of endotrivial representations. Our description involves functions that we call weak -homomorphisms. These are generalizations to possibly non-normal of the classical homomorphisms appearing in the normal case.
Balogh, Zsolt (2004)
Acta Mathematica Academiae Paedagogicae Nyí regyháziensis. New Series [electronic only]
Huerta-Aparicio, Luis, Molina-Rueda, Ariel, Raggi-Cárdenas, Alberto, Valero-Elizondo, Luis (2009)
Revista Colombiana de Matemáticas
Hans-Werner Henn, Stewart Priddy (1994)
Commentarii mathematici Helvetici
Brent Kerby, Jonathan D. H. Smith (2010)
Commentationes Mathematicae Universitatis Carolinae
The automorphisms of a quasigroup or Latin square are permutations of the set of entries of the square, and thus belong to conjugacy classes in symmetric groups. These conjugacy classes may be recognized as being annihilated by symmetric group class functions that belong to a -ideal of the special -ring of symmetric group class functions.
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