Exponent two implies abelian

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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

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.