Equivalence of subgroups

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Suppose G_1 and G_2 are groups, with H_1 a subgroup of G_1 and H_2 a subgroup of G_2. An equivalence of subgroups between the group-subgroup pairs H_1 \le G_1 and H_2 \le G_2 is an isomorphism of groups \sigma:G_1 \to G_2 such that the restriction of \sigma to H_1 defines an isomorphism from H_1 to H_2.