Hi,

Can anyone help with the following please?

A non-Abelian group can have both Abelian and non-Abelian subgroups. The symmetric groups Sn are non-Abelian for n ≥ 3.

a. Prove that the alternating group, An, which is a subgroup of Sn, is non-Abelian for n ≥ 3.

b. Construct an Abelian and a non-Abelian subgroup of S4 (excluding the trivial cases G = {e} and G = S4).

For a) i was thinking i need to find two subgroups say (1 2 3...) and (1 3 2...) (in cycle notation) and show that they are not commutative but (1 2 3)(1 3 2) = (1)(2)(3)... and (1 3 2) (1 2 3) = (1)(2)(3)...so this example is commutative but yields a result which is not an element of A4 (the set of even permutations). Am I on the right lines with this?

b) I'm unsure of but solving a) would help I assume.

Thanks in advance.