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 formula_2 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