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

## Source Work

Chapter $13$: Direct products:
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:
$x y x^{-1} = y x^4 x^{-1} = y x^3 \in N$