.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
Almost correct but not quite: , 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 are conjugated iff they've the same length. This also answers your question below.
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.