Singleton is Linearly Independent

Theorem
Let $$\left({G, +_G}\right)$$ be a group whose identity is $$e$$.

Let $$\left({G, +_G: \circ}\right)_K$$ be a $K$-vector space.

Let $$x \in G: x \ne e$$.

Then $$\left\{{x}\right\}$$ is a linearly independent subset of $$G$$.

Proof
Follows directly from Basic Vector Results and Basic Results about Modules.