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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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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