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| In algebra, more specifically [[group theory]], a ''p''-'''elementary group''' is a [[direct product of groups|direct product]] of a finite [[cyclic group]] of order relatively prime to ''p'' and a [[p-group|''p''-group]]. A finite group is an elementary group if it is ''p''-elementary for some prime number ''p''. An elementary group is [[nilpotent group|nilpotent]].
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| [[Brauer's theorem on induced characters]] states that a character on a finite group is a linear combination with integer coefficients of characters [[induced character|induced]] from elementary subgroups.
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| More generally, a finite group ''G'' is called a ''p''-'''hyperelementary''' if it has the extension
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| :<math>1 \longrightarrow C \longrightarrow G \longrightarrow P \longrightarrow 1</math>
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| where <math>C</math> is cyclic of order prime to ''p'' and ''P'' is a ''p''-group. Not every hyperelementary group is elementary: for instance the non-abelian group of order 6 is 2-hyperelementary, but not 2-elementary. The term "hyperelementary" was introduced {{Citation needed|date=March 2011}} by G. Segal.
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| ==References==
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| * Arthur Bartels, Wolfgang Lück, ''[http://wwwmath.uni-muenster.de/u/bartelsa/research/induction.pdf Induction Theorems and Isomorphism Conjectures for K- and L-Theory'']
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| * G. Segal, ''[http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1968__34_/PMIHES_1968__34__113_0/PMIHES_1968__34__113_0.pdf The representation-ring of a compact Lie group]''
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| * J.P. Serre, "Linear representations of finite groups". Graduate Texts in Mathematics, vol. 42, Springer-Verlag, New York, Heidelberg, Berlin, 1977,
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| [[Category:Representation theory of finite groups]]
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| {{Abstract-algebra-stub}}
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My name is Emmanuel and I am studying Architecture, Art, and Planning and Biology at Rinn / Austria.
Review my page - commercial roofing perth