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Second isomorphism theorem

Statement

Given a group GG and subgroups H⊆GH \subseteq G, N⊲GN \triangleleft G, then there is an isomorphism HNN≅HH∩N\frac{HN}{N} \cong \frac{H}{H \cap N}

Proof

(show/hide)

We have that HN≤GHN \leqslant G, and H∩N≤HH \cap N \leqslant H. (see here)

Compose the map ϕ:H→HNN\phi: H \to \frac{HN}{N} from:

  1. inclusion H→HNH \to HN - kernel is zero
  2. quotient map HN→HNNHN \to \frac{HN}{N} - kernel is NN

We have ker⁡(ϕ)=H∩N\ker(\phi)=H \cap N, so the statement follows from the first isomorphism theorem.

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all these pages adapted with probably insufficient credit from my university's lecture notes