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Correspondence Theorem

  • Definition: Let f: G -> G' be a surjective group homomorphism with kernel K. There is a bijective correspondence between subgroups of G' and subgroups of G that contain K: {subgroups of G that contain K} <--> {subgroups of G'}. This correspondence is defined as follows: a subgroup H of G that contains K maps its image f(H) in G', a subgroup H' of G' maps its inverse image f^(-1)(H') in G. If H and H' are corresponding subgroups, then H is normal in G if and only if H' is normal in G'. If H and H' are corresponding subgroups, then |H| = |H'||K|.

    Source: Algebra, second edition by Michael Artin

  • Example: A similar theorem exists for the correspondence between the ideals of two rings in a surjective ring homomorphism.

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