Rational and abelian implies elementary abelian 2-group

From Groupprops
Jump to: navigation, search


Suppose G is a group that is both a rational group and an abelian group. Then, G must be an elementary abelian 2-group, i.e., an elementary abelian 2-group, i.e., it is isomorphic to the additive group of a vector space over field:F2, or equivalently, it is an abelian group of exponent dividing 2.

Related facts


This follows quite directly: rational implies that every element is conjugate to its inverse, which in the abelian case forces every element to be equal to its inverse, thus forcing all elements to have order dividing 2.