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Algorithmic questions concerning isomorphism testing. A prime example is the graph isomorphism problem, which is to decide if two input graphs are isomorphic. This tag may also be used for isomorphism testing between other objects (such as groups).
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Finite objects for which isomorphism is NP-hard or harder?
Perhaps this is not exactly what you meant, but isomorphism testing of succinct graphs is known to be hard for PSPACE; that is, succinct graphs are ones given implicitly by a circuit $C$ where $C(u,v) …