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In group theory, conjugacy is the relation between elements of a group that states that one element is the conjugate of the other. This relation is an equivalence relation, and the equivalence classes are the conjugacy classes of the group.

Another way of stating this is to say that conjugation is group action of G on itself, and the conjugacy classes are the orbits of this action.

The conjugacy problem is the decision problem of determining from a presentation of a group whether two elements of the group are conjugate .

The conjugacy problem was identified by Max Dehn in 1911 as one of three fundamental decision problems in group theory; the other two being the group isomorphism problem and the word problem.