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.

The only sequence of distinct terms in $$\left\{{x}\right\}$$ is the one that goes: $$x$$.

So the only way to make $$\sum_{k=1}^1 \lambda_k \circ a_k = e$$ is to make $$\lambda_1 = 0_R$$.