Singleton is Linearly Independent
Let $K$ be a division ring.
Let $x \in G: x \ne e$.
Then $\set x$ is a linearly independent subset of $G$.
The only sequence of distinct terms in $\set x$ is the one that goes: $x$.
Suppose $\exists \lambda \in K: \lambda \circ x = e$.
From Zero Vector Space Product iff Factor is Zero it follows that $\lambda = 0$.
Hence the result from definition of linearly independent set.