# Group of prime exponent

From Groupprops

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 is a prime number, a group of exponent is a (nontrivial) group in which every element has order .

## Particular cases

Value of prime ? | What can we say about groups of exponent |
---|---|

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 , it must be a quotient of the Burnside group , which is a finite group of size |

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 for which the Burnside problem has an answer of *No*, it is possible to have an infinite group of exponent with a finite generating set. However, there will still be many finite groups of interest with exponent and a finite generating set.