/sci/ - Science & Math

I have a very stupid Algebra question.
Let [math]G = S_4 [/math] be the symmetric group and [math]H = \{(123),(132),(1)\}[/math] be a subgroup of [math]G[/math]. Further let [math] \sigma = (14)[/math].
The problem asks to calculate [math]\sigma H [/math] and [math] H \sigma [/math] and then asks whether [math]H[/math] is a normal subgroup of [math]G[/math].

What I calculated is
[eqn] \sigma H = \{(1234),(1324),(14)\} \\
H \sigma = \{(1423),(1432),(14)\} [/eqn]
since they are different, I concluded that [math]H [/math] is not a normal subgroup.

But the book claims that
[eqn] \sigma H = \{(423),(432),(14)\} \\
H \sigma = \{(423),(432),(14)\} [/eqn]
and that [math]H [/math] is a normal subgroup of [math]G[/math].

What is going on here?