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