P-group

Definition

Let $p$ be a prime. A $p$-group is a group in which every element has finite order, and the order of every element is a power of $p$.

The term $p$-group is typically used for a finite $p$-group, which is equivalent to a group of prime power order.

Relation with other properties

Stronger properties

• Group of prime power order
• Nilpotent p-group: For proof of the implication, refer and for proof of its strictness (i.e. the reverse implication being false) refer p-group not implies nilpotent.

Weaker properties

• Periodic group

See also

• 2-group