Isomorphism Theorem - Groups - Statement of The Theorems - Second Isomorphism Theorem

Second Isomorphism Theorem

Let G be a group. Let S be a subgroup of G, and let N be a normal subgroup of G. Then:

  1. The product SN is a subgroup of G,
  2. The intersection S ∩ N is a normal subgroup of S, and
  3. The quotient groups (SN) / N and S / (S ∩ N) are isomorphic.

Technically, it is not necessary for N to be a normal subgroup, as long as S is a subgroup of the normalizer of N. In this case, the intersection S ∩ N is not a normal subgroup of G, but it is still a normal subgroup of S.

Read more about this topic:  Isomorphism Theorem, Groups, Statement of The Theorems

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