# Math Help - Isometries

1. ## Isometries

Show that $D_4$ (isometries of a square with vertices 1,2,3,4) is not normal in $S_4$

(Permutation of {1,2,3,4})
$|D_4|=8$
$|S_4|=4!$

Any help is appreciated! Thanks!

2. Originally Posted by Linnus
Show that $D_4$ (isometries of a square with vertices 1,2,3,4) is not normal in $S_4$

(Permutation of {1,2,3,4})
$|D_4|=8$
$|S_4|=4!$

Any help is appreciated! Thanks!
Identify $D_4$ with a definite subgroup of $S_4$ , since you want to show it is not normal there: it will contain two 4-cycles. Now conjugate one of these 4-cycles by a nicely chosen transposition and get a new 4-cycle that isn't contained in your $D_4$ ...

Another way: $D_4$ is a Sylow 2-subgroup of $S_4$ so it is normal there iff there's one unique subgroup of its order (8); Well, show that there are more than one...

Tonio