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Elementary group
In algebra, more specifically group theory, a ''p''-elementary group is a direct product of a finite cyclic group of order relatively prime to ''p'' and a ''p''-group. A finite group is an elementary group if it is ''p''-elementary for some prime number ''p''. An elementary group is nilpotent.
Brauer's theorem on induced characters states that a character on a finite group is a linear combination with integer coefficients of characters induced from elementary subgroups.
More generally, a finite group ''G'' is called a ''p''-hyperelementary if it has the extension
:
where 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.
==See also==
Elementary abelian group
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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