Second isomorphism theorem

Algebra Level 4

Let \(S_3\) be the symmetric group on three letters. Let \(H\) be the subgroup generated by the transposition \( (12),\) and let \(N\) be the subgroup generated by the transposition \( (13).\)

Then the second isomorphism theorem gives an isomorphism \[HN/N \simeq H/(H \cap N),\] where \(HN = \{ hn : h \in H, n \in N\}.\)

Now \(H \cap N = \{1\},\) so the right side has order \(2.\) So the left side has order \(2.\) Now \(|N|=2\) and \(|HN/N|=2,\) so \(|HN|=4.\) But Lagrange's theorem says that \(|HN|\) must divide \(|S_3|,\) which is \(6.\)

What has gone wrong with this argument?

I. As defined above, \(HN\) is not a subgroup of \(S_3,\) so Lagrange's theorem does not apply.
II. The order of a quotient \(G/K\) of finite groups is not always equal to \(|G|/|K|.\)
III. \(N\) is not normal, so the second isomorphism theorem does not apply.

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