I am having trouble with this problem. We just started learning it and I am unsure of how to obtain all the elements and how to form them into cyclic subgroups.
The alternating group A4 consists of the identity together with eight 3-cycles and three pairs of 2-cycles. Each 3-cycle generates a cyclic subgroup H of order 3. Each pair of 2-cycles also generates a cyclic subgroup K of order 2.
Write down all 12 elements and arrange them into their cyclic subgroups. Pick any two elements from two separate 3-cycles (e.g. α= (123) and β = (13)(24) and show you can generate all of A4 from these two elements. Express all 12 elements as products of ↵ and .
Show that the 3-cycle α = (123) together with the 2-cycle β = (13)(24) also generates all of A4.
What do you generate from the elements (13)(24) and (12)(34)? Is it the entire group or is it a subgroup?
Carefully draw a subgroup lattice diagram for A4.
$\endgroup$ 11 Answer
$\begingroup$Using the answers of this MSE-question, we see that the subgroup lattice of the alternating group $A_4$ is as follows:
We also see, that the group generated by $(12(34)$ and $(13)(24)$ is not the full group.
$\endgroup$ 2
