# One and Minus One form Subgroup of Multiplicative Group of Rational Numbers

## Theorem

Let $\struct {\Q_{\ne 0}, \times}$ be the multiplicative group of rational numbers.

Let $S \subseteq \Q$ where $S = \set {1, -1}$.

Then $\struct {S, \times}$ is a subgroup of $\struct {\Q_{\ne 0}, \times}$.

## Proof

By hypothesis, $S$ is not empty.

As $0 \notin S$, it follows that $S \subseteq \Q_{\ne 0}$.

Recall that $-1 \times -1 = 1$ and also $1 \times 1 = 1$.

Thus:

$\forall x \in S: x \times y^{-1} \in S$

The result follows from the One-Step Subgroup Test.

$\blacksquare$