Thread: Isometries

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

(Permutation of {1,2,3,4})



Any help is appreciated! Thanks!

Originally Posted by Linnus

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

(Permutation of {1,2,3,4})



Any help is appreciated! Thanks!

Identify with a definite subgroup of , 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 ...

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

Tonio