Order is Preserved by Group Isomorphism
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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