# Normal Cyclic Subgroup

## Homework Statement

Is the Cyclic Subgroup { (1), (123), (132)} normal in $$A_{4}$$ (alternating group of 4)

## The Attempt at a Solution

So I believe if I just check if gH=Hg for all g in A_4 that would be suffice to show that it is a normal subgroup, but that seems really tedious. Is there a easier way?

Also how can I figure out what the elements of A_4 are? I know its the even permutations but is there a way to quickly identity which ones it is? How do I visualize it?

Visually speaking, A_4 is the group associated with rotations of the regular tetrahedron, if that helps.

One can use the Sylow theorems to prove normality sometimes, but in this case it doesn't help.

So only way is by brute force?

Dick