Normal Cyclic Subgroup in A_4: Proving Normality and Identifying Elements

[polarbears]

Homework Statement

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

Homework Equations

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?

 

[e(ho0n3]

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.

 

[polarbears]

So only way is by brute force?

 

[Dick]

So only way is by brute force?

No, it's not the only way. In this case, I would try and guess an element g of A4 such that gHg^(-1) is NOT equal to H, where H is your subgroup. It's not hard. You can recognize whether a permutation is even just by looking at it's cycle structure. Cycles with an odd number of elements are even permutations and cycles with an even number of elements are odd permutations.