Elementary Group
Get Elementary Group essential facts below. View Videos or join the Elementary Group discussion. Add Elementary Group to your PopFlock.com topic list for future reference or share this resource on social media.
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

${\displaystyle 1\longrightarrow C\longrightarrow G\longrightarrow P\longrightarrow 1}$

where ${\displaystyle C}$ 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.