Elementary group

In finite-group character theory, an elementary group is a group that is p-elementary group for some prime $p$. Such groups are solvable and provide the subgroup tests in Brauer's characterization of characters. This does not mean elementary abelian group.

Write an elementary group as $P\times C$ with $P$ a p-group and $C$ cyclic of order prime to $p$. Split $C$ into its cyclic pi-part and pi-complementary part. Attach $P$ to whichever part contains $p$, producing a direct product of groups of a pi-group and a complementary-prime group.

A p-elementary group is a direct product of groups of a p-group and a cyclic group whose order is coprime to $p$. Either factor may be trivial. The coprime cyclic factor is central, a stronger requirement than merely having a cyclic subgroup with p-group quotient.