= Group extension by a duplicate carrier
{title2=$A\times C_2$}
If a <group> has carrier $A$ and a disjoint <set> $B$ is in <bijection> with $A$, identify $A\cup B$ with $A\times C_2$, sending the original carrier to $A\times\{0\}$. Transporting the direct-product multiplication gives a larger <group operation> that agrees exactly with the old one on $A$. A maximal partial <group> on a fixed carrier universe therefore has complement of <cardinality> strictly below its own, using cardinal comparability.
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