# 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 $yxy^{-1}=x^{i}$ for some $i<7$ since $K$ is normal. Now $x=y^{3}xy^{-3}=y^{2}x^{i}y^{-2}=yx^{i^{2}}y^{-1}=x^{i^{3}}$, or $i^{3}=1\mod 7$. This implies $i=1,2$, or 4.

Case 1: $yxy^{-1}=x$, so $G$ is abelian and isomorphic to $C_{21}=C_{3}\times C_{7}$.

Case 2: $yxy^{-1}=x^{2}$, then every product of the elements $x,y$ can be reduced to one in the form $x^{i}y^{j}$, $0\leq i<7$, $0\leq j<3$. These 21 elements are clearly distinct, so $G=\langle x,y\mid x^{7}=y^{3}=1,yx=x^{2}y\rangle$.

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

Title example of groups of order pq ExampleOfGroupsOfOrderPq 2013-03-22 14:51:15 2013-03-22 14:51:15 jh (7326) jh (7326) 4 jh (7326) Example msc 20D20