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.

$\blacksquare$

Sources

• 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 4$. Vector Spaces
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases