I'm working on building a list of the normal subgroups of order 4 of
I know that all subgroups of order 4 are either isomorphic to or
I want to show that any subgroup of order 4 that is isomorphic to will not be normal.
Since is cyclic, the subgroups in question must also be cyclic. So these groups must be generated by a permutation of of length 4.
For example, consider
I want to show that such that
For ease of calculation I use
We see that
So H is not normal
I want to generalize this for any cyclic subgroup of 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.