Exponent two implies abelian

Statement

Any group whose exponent is two must be abelian. In fact, it is an elementary abelian 2-group, or, more precisely, the additive group of a vector space over the field of two elements.

Conversely, the additive group of any nonzero vector space over the field of two elements is a group of exponent two.

Related facts

Breakdown for higher exponents

It is possible to have a group of prime exponent for any odd prime that is non-abelian. The standard example is the unitriangular matrix group:UT(3,p) which is a non-abelian group of order $p^{3}$ and exponent $p$. Also, it is possible to have non-abelian groups whose exponent is a higher power of $2$. For instance, the dihedral group of order eight has exponent four.

Other related facts

• Exponent three implies class three (follows from exponent three implies 2-Engel for groups and 2-Engel implies class three for groups)
• Exponent five not implies nilpotent
• See also the Burnside problem, which seeks to determine the freest possible groups of a given exponent.

Facts used

1. Square map is endomorphism iff abelian

Proof

The proof follows from fact (1), along with the observation that in a group of exponent two, the square map is the trivial map and is thus an endomorphism.