Discussion
Which ONE of the following statements is FALSE?
Answer: D
The FALSE statement is \(\mathbb{Z}_{6}\) is isomorphic to a quotient group of \(S_{4}\).
The normal subgroups of \(S_{4}\) are \(\{e\}\), the Klein four-group \(V_4\), \(A_4\) and \(S_4\). Their quotients have orders 24, 6, 2 and 1, and are \(S_4\), \(S_4/V_4 \cong S_3\), \(\mathbb{Z}_2\) and the trivial group. The only quotient of order 6 is \(S_3\), which is non-abelian, so the cyclic group \(\mathbb{Z}_6\) is not a quotient.
The other statements are true. The permutations fixing the symbol 4 form a subgroup isomorphic to \(S_3\). The 3-cycle \((1\,2\,3)\) generates a subgroup isomorphic to \(\mathbb{Z}_3\). And \(S_4/V_4 \cong S_3\), as above, so \(S_3\) is also a quotient.
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