A4 Subgroup Question: Why is it Normal?

[tyrannosaurus]

Homework Statement

There is only one subgroup of order 4 in A4 (Alternating group of degree 4)(This subgroup is (1), (12)(34), (13)(24), (14)(23)). Why does this imply that this subgroup must be a normal subgroup in A4? Generalize to arbitrary finite groups.

Homework Equations

The Attempt at a Solution

I thought about using the fact that for some g that is an element of a group G, |gH^-1g^-1|=|H| (the order of the subgroup H equals each of its conjugates orders) in the whole group, I am not sure were to go from here. Can some help me on this one at all?

 

[mighty2000]

Thank you for bringing up this interesting topic. The fact that there is only one subgroup of order 4 in A4 implies that it must be a normal subgroup because it is the only subgroup of its size and therefore must be invariant under any automorphism of A4. In other words, any element g in A4 will not change the structure of this subgroup, since it is the only one of its size. This is why it is considered a normal subgroup.

To generalize this to arbitrary finite groups, we can use the same logic. If there is only one subgroup of a certain order in a finite group, then it must be a normal subgroup because it is the only one of its size and therefore must be invariant under any automorphism of the group. This holds true for any finite group, not just A4.

I hope this helps clarify the concept of normal subgroups in finite groups. If you have any further questions, please don't hesitate to ask. Keep up the great work in your studies!