example of groups of order pq

As a specific example, let us classify groups of order 21. Let $G$ be a group of order 21. There is only one Sylow 7-subgroup $K$ so it is normal. The possibility of there being conjugate Sylow 3-subgroups is not ruled out. Let $x$ denote a generator for $K$, and $y$ a generator for one of the Sylow 3-subgroups $H$. Then $x^7=y^3=1$, and $yx{y}^{-1}=x^i$ for some since $K$ is normal. Now $x=y^{3}x{y}^{-3}=y^2{x}^i{y}^{-2}=y{x}^{i^2}{y}^{-1}=x^{i^3}$, or $i^3=1mod7$. This implies $i=1,2$, or 4.

Case 1: $yx{y}^{-1}=x$, so $G$ is abelian and isomorphic to$C_{21}=C_3\times C_7$.

Case 2: $yx{y}^{-1}=x^2$, then every product of the elements $x,y$can be reduced to one in the form $x^i{y}^j$, , . These 21 elements are clearly distinct, so $G=⟨x,y\mid x^7=y^3=1,yx=x^{2}y⟩$.

Case 3: $yx{y}^{-1}=x^4$, then since $y^2$ is also a generator of$H$ and $y^{2}x{y}^{-2}=y{x}^4{y}^{-1}=x^{16}=x^2$, we have recoveredcase 2 above.