# Order is Preserved by Group Isomorphism

## Theorem

Let $G$ and $H$ be groups.

Let $\phi: G \to H$ be a (group) isomorphism.

Then:

- $\order G = \order H$

where $\order {\, \cdot \,}$ denotes the order of a group.

## Proof

By definition, an isomorphism is a bijection.

By definition, the order of a group is the cardinality of its underlying set.

The result follows by definition of set equivalence.

$\blacksquare$

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 7.3$. Isomorphism