How to find the elements of a subgroup?

[golmschenk]

If you have a subgroup and it's order how do you find the elements of the group? I'd be happy with any example to help explain this, but just so there's something to go off of my example would be if you have a subgroup A₄ = <(1 2 3), (1 2)(3 4)> of S₄ that has an order of 12 how do you find it's elements? Thanks for your time!

 

[Tinyboss]

Just knowing the order won't tell you the elements in general; for instance, Q8 (the quaternions) has three different order-four subgroups, <i>, <j>, and <k>. So if all you're given is a group and the order of a subgroup, and you're expected to find the elements, then it must be the case that only one subgroup of the given order exists.

 

[rs1n]

If you have a subgroup and it's order how do you find the elements of the group? I'd be happy with any example to help explain this, but just so there's something to go off of my example would be if you have a subgroup A₄ = <(1 2 3), (1 2)(3 4)> of S₄ that has an order of 12 how do you find it's elements? Thanks for your time!

There is no general way to do this as far as I know. However, there are many special cases where you can determine all the elements. In particular, A4 is the subgroup of even cycles, so you can just write down the 12 even cycles of S4. Other techniques:

1) If your group G is cyclic, this is trivial since G = { 1, x, x^2, x^3, ... }

2) If G is a group of order pq (where p and q are primes), and G has only 1 subgroup H of order p and only 1 subgroup K of order q, and the intersection of H and K is just the identity, then G is a cyclic group.

etc.

Basically, it boils down to whether or not you have a special case of certain types of groups. I have only barely scratched the surface with the two cases listed above. Generally, there really is no general method (I know of) for listing all elements of a group when only given a group G and one subgroup H.