Math Help - Subgroups of order 4 of S4

1. Subgroups of order 4 of S4

I'm working on building a list of the normal subgroups of order 4 of $S_4$
I know that all subgroups of order 4 are either isomorphic to $\mathbb{Z}_4$ or $\mathbb{Z}_2 \oplus \mathbb{Z}_2$

I want to show that any subgroup of order 4 that is isomorphic to $\mathbb{Z}_4$ will not be normal.

Since $\mathbb{Z}_4$ is cyclic, the subgroups in question must also be cyclic. So these groups must be generated by a permutation of $S_4$ of length 4.

For example, consider $H = <(1234)>$
I want to show that $\exists g \in S_4$ such that $g(1234)g^{-1} \notin H.$
For ease of calculation I use $g = (12).$
We see that $H = \{ e, (1234), (13)(24), (1432) \}$
yet $g(1234)g^{-1} = (134) \notin S_4$
So H is not normal

I want to generalize this for any cyclic subgroup of $S_4$ but I cannot think of a way other than brute force to do this. I only want a hint on this, a push in the right direction.

2. Originally Posted by Haven
I'm working on building a list of the normal subgroups of order 4 of $S_4$
I know that all subgroups of order 4 are either isomorphic to $\mathbb{Z}_4$ or $\mathbb{Z}_2 \oplus \mathbb{Z}_2$

I want to show that any subgroup of order 4 that is isomorphic to $\mathbb{Z}_4$ will not be normal.

Since $\mathbb{Z}_4$ is cyclic, the subgroups in question must also be cyclic. So these groups must be generated by a permutation of $S_4$ of length 4.

For example, consider $H = <(1234)>$
I want to show that $\exists g \in S_4$ such that $g(1234)g^{-1} \notin H.$
For ease of calculation I use $g = (12).$
We see that $H = \{ e, (1234), (13)(24), (1432) \}$
yet $g(1234)g^{-1} = (134) \notin S_4$
So H is not normal

Almost correct but not quite: $g(1234)g^{-1}=(1342)\notin <(1234)>$ , so you still get that the subgroup isn't normal.
Remember that conjugating ANY cycle by ANY permutation gives another cycle of EXACTLY the same length as the original one, and also that any two cycles of the same length in $S_n$ are conjugated iff they've the same length. This also answers your question below.

Tonio

I want to generalize this for any cyclic subgroup of $S_4$ but I cannot think of a way other than brute force to do this. I only want a hint on this, a push in the right direction.
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