= Solution
<Zorn's lemma> states that a nonempty <partially ordered set> in which every chain has an <upper bound> has a <maximal element>. Order the partial <group> operations by inclusion of their graphs. The collection is a <set>, since its members are <subsets> of $X\times X\times X$. It is nonempty: choose $x_0\in X$ and put the one-element <group operation> on $\{x_0\}$.
To construct a <chain union of partial group operations>, for the given nonempty chain, let $m=\bigcup_{n\in\mathcal C}n$ and let $A$ be the union of the corresponding <group> carriers. Comparability guarantees agreement wherever two operations are both defined, so $m$ is a function. Its domain is $A\times A$: any two elements of $A$ lie together in one carrier, by taking the larger of the two chain members containing them. Their product then lies in that carrier and hence in $A$.
Any three elements similarly lie together in one chain member, where <associativity> holds; therefore <associativity> holds for $m$. The identities of nested nonempty <groups> agree: the smaller identity remains an <idempotent> under the larger <group operation>, whose only <idempotent> is its identity. Hence the whole chain has a common identity. Each element's inverse in a carrier remains its inverse in every larger carrier, so it is an inverse in the union as well. Thus $(A,m)$ satisfies all <group axioms> and
$$
\boxed{\bigcup\mathcal C\in\mathcal G.}
$$
It is an <upper bound> for the chain. The empty chain also has an <upper bound> because $\mathcal G$ is nonempty, completing the hypothesis needed for <Zorn's lemma>.
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