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