Group of prime exponent

The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter
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Definition

A group of prime exponent is a group whose exponent is a prime number. If $p$ is a prime number, a group of exponent $p$ is a (nontrivial) group in which every element has order $p$.

Particular cases

Value of prime $p$?	What can we say about groups of exponent $p$
2	must be elementary abelian. See exponent two implies abelian
3	must be 2-Engel and class three. Also, if it has a generating set of finite size $m$, it must be a quotient of the Burnside group $B(m,3)$, which is a finite group of size $3^{m+{\binom {m}{2}}+{\binom {m}{3}}}$
5	no bound on nilpotency class. Unknown whether finite generating set forces the group to be finite.

For related information, see the Burnside problem. Note that for those primes $p$ for which the Burnside problem has an answer of No, it is possible to have an infinite group of exponent $p$ with a finite generating set. However, there will still be many finite groups of interest with exponent $p$ and a finite generating set.

Relation with other properties

Stronger properties

• Elementary Abelian group