Complement to normal subgroup is isomorphic to quotient group

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Statement

Suppose G is a group, N is a normal subgroup and H is a subgroup such that N and H are permutable complements: NH = G and N \cap H is trivial. Then, G/N \cong H.

In particular, any two permutable complements to N are isomorphic to each other.

Related facts

Caveats

Note the following:

Other related facts

Proof

Proof using second isomorphism theorem

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