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 whose zero is $0_K$.

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

Then $\left\{{x}\right\}$ is a linearly independent subset of $G$.

Proof

The only sequence of distinct terms in $\left\{{x}\right\}$ 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

• Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.4$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 27$