Group extension by a duplicate carrier ($A\times C_2$)

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.