Groups of Order 30/C 3 x D 5/Mistake

Source Work

 * Chapter $13$: Direct products:
 * Exercise $2$ (solution)
 * Exercise $2$ (solution)

Mistake

 * Now let $N$ be the subgroup of $G$ generated by $x^3$ and $y$. Note that since
 * $y x^3 y^{-1} = \paren {y x y^{-1} }^3 = \paren {x^4}^3 = x^{12} = x^{-3}$,
 * $N$ is isomorphic to the dihedral group $\map D 3$. ...


 * ... To show that $x$ normalises $N$, note that
 * $x x^3 x^{-1} = x^3$; and $x y x^{-1} = y x^4 x^{-1} = x y^3 \in N$.

Correction
There are two mistakes here:


 * $(1): \quad$ In the first section, it should say:
 * $N$ is isomorphic to the dihedral group $\map D 5$.


 * $(2): \quad$ That second expression should read:
 * $x y x^{-1} = y x^4 x^{-1} = y x^3 \in N$