Second Isomorphism Theorem for Groups

Second Isomorphism Theorem for Groups Let \( G \) be a group, \( H \leq G \) and \( N \unlhd G \). Then

1. \( HN \leq G \)
2. \( H \cap N \unlhd H \)
3. \( HN / N \cong H / ( H \cap N ) \)

Proof.

1. \( N \unlhd G \implies HN=NH \implies HN \leq G \)
2. \( \forall x \in H \cap N, x \in H \land x \in N. \) Since \( N \unlhd G \), \( \forall x \in H \cap N; h \in H, hxh^{-1} \in N \). Note, \( x,h,h^{-1} \in H \). So, \( hxh^{-1} \in H \). Thus, \( hxh^{-1} \in H \cap N \) and \( H \cap N \unlhd H \).
3. Consider now the canonical homomorphism \( \varphi : HN \longrightarrow HN / N \) by \( \varphi(h_in)=h_iN \)

   The restriction of \( \varphi \) to \( H \) is a homomorphism of \( H \) in \( HN / N \) with kernel: \( H \cap \operatorname { ker } ( \varphi )=H \cap N \). It is not difficult to see that \( \operatorname { im } ( \varphi_{H} )=HN/N \). Based on First Isomorphism Theorem, \( H / ( H \cap N ) \cong HN / N \)

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