# Singleton is Linearly Independent

## Theorem

Let $K$ be a division ring.

Let $\struct {G, +_G}$ be a group whose identity is $e$.

Let $\struct {G, +_G, \circ}_K$ be a $K$-vector space whose zero is $0_K$.

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

Then $\set x$ is a linearly independent subset of $G$.

## Proof

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.

$\blacksquare$