Prove that every group of order \(6\) is isomorphic to either \[ \ZZ/6\ZZ \qquad\text{or}\qquad S_3. \]
Solution
Let \(G\) be a group of order \(6\).
1. There are elements \(a,b\in G\) such that \(a\) has order \(3\), \(b\) has order \(2\), \(P\coloneqq\langle a\rangle\) is normal in \(G\), and every element of \(G\) has a unique expression \(a^i b^j\) with \(0\leq i<3\) and \(0\leq j<2\).
By Sylow’s theorems, the number \(n_3\) of Sylow \(3\)-subgroups satisfies \[ n_3\equiv1\pmod3, \qquad n_3\mid2, \] so \(n_3=1\). Thus the Sylow \(3\)-subgroup \(P\) is normal and cyclic; write \(P=\langle a\rangle\). By Cauchy’s theorem there is an element \(b\) of order \(2\). Then \(P\cap\langle b\rangle=\{1\}\), and \[ \abs{P\langle b\rangle} =\frac{\abs P\,\abs{\langle b\rangle}}{\abs{P\cap\langle b\rangle}} =6. \] Hence \(G=P\langle b\rangle\). The trivial intersection also makes the expression \(a^i b^j\) unique.
2. Exactly one of the relations\[ bab^{-1}=a \qquad\text{or}\qquad bab^{-1}=a^{-1} \]holds.
Because \(P\) is normal, conjugation by \(b\) restricts to an automorphism of the cyclic group \(P\) of order \(3\). An automorphism sends the generator \(a\) to either \(a\) or \(a^2=a^{-1}\).
3. If \(bab^{-1}=a\), then \(G\cong\ZZ/6\ZZ\).
In this case \(a\) and \(b\) commute. If \((ab)^k=1\), then \[ a^k=b^{-k}\in P\cap\langle b\rangle=\{1\}. \] Thus \(3\mid k\) and \(2\mid k\), so \(6\mid k\). Therefore \(ab\) has order \(6\) and generates \(G\), proving that \(G\) is cyclic of order \(6\).
4. If \(bab^{-1}=a^{-1}\), then \(G\cong S_3\).
Let \[ r=(123), \qquad s=(12) \] in \(S_3\). They satisfy \(r^3=s^2=1\) and \(srs^{-1}=r^{-1}\). By step 1, every element of \(G\) has a unique normal form \(a^i b^j\), and the relation \(ba=a^{-1}b\) determines multiplication of these normal forms. The same multiplication rule holds for the six elements \(r^i s^j\) of \(S_3\). Hence \[ a^i b^j\longmapsto r^i s^j \] is a well-defined bijective homomorphism \(G\to S_3\).
5. Q.E.D.
Step 2 exhausts the possible conjugation actions, and steps 3 and 4 give the corresponding isomorphism types.