P-elementary group

Symbol-free definition
Let $$p$$ be a prime. A finite group is termed $$p$$-elementary if it is the direct product of a $$p$$-group and a cyclic group of order relatively prime to $$p$$.

A group is termed elementary if it is $$p$$-elementary for some prime $$p$$.

Definition with symbols
Let $$p$$ be a prime. A finite group $$G$$ is termed $$p$$-elementary if $$G$$ is the internal direct product of a $$p$$-subgroup $$P$$ and a cyclic subgroup $$C$$ whose order is relatively prime to $$p$$.

A group is termed elementary if it is $$p$$-elementary for some prime $$p$$.

Relation with other properties
Elementary groups are nilpotent. This is because cyclic groups are nilpotent, and $$p$$-groups are also nilpotent, and a product of nilpotent groups is nilpotent.