Second isomorphism theorem

Algebra Level 2

Let S3S_3 be the symmetric group on three letters. Let HH be the subgroup generated by the transposition (12), (12), and let NN be the subgroup generated by the transposition (13). (13).

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

Now HN={1},H \cap N = \{1\}, so the right side has order 2.2. So the left side has order 2.2. Now N=2|N|=2 and HN/N=2,|HN/N|=2, so HN=4.|HN|=4. But Lagrange's theorem says that HN|HN| must divide S3,|S_3|, which is 6.6.

What has gone wrong with this argument?

I. As defined above, HNHN is not a subgroup of S3,S_3, so Lagrange's theorem does not apply.
II. The order of a quotient G/KG/K of finite groups is not always equal to G/K.|G|/|K|.
III. NN is not normal, so the second isomorphism theorem does not apply.

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