Let $G $ be an abelian group. In this section we shall use additive notation for $G$, so the group operation will be denoted by ' $+$ ' the identity by $0$, the inverse of $g$ by $-g$ and powers of $g$ by $2g, 3g, ....$
Let $2G$ be the subgroup of $G$ consisting of all elements of the form $g + g$ $(g \in G)$. If $G$ has a basis of $n$ elements, then $G/2G$ is a group of order $2^n$.
How the order of $G/2G$ is $2^n$? Plz show in an elementary way.
This an excerpt from "Algebraic number theory and Fermats last theorem by Ian Stewart, David Tall (3rd ed.).