# Second isomorphism theorem

Algebra Level 2

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.

×