Order of a finite group of exponent two
= Order of a finite group of exponent two
{title2=$|G|=2^d$}
A <group of exponent two is abelian>. Give such a finite group the additive notation of a <vector space> over $\mathbb F_2$, with zero acting as zero and one as the identity scalar. The group axioms and $g+g=0$ verify the vector-space laws. A <basis> with $d$ vectors then gives exactly $2^d$ elements. Thus a finite group whose elements all have order at most two must have power-of-two order.