Finding 2-Sylow & 3-Sylow of ##S_4##

[Lee33]

Homework Statement

Find the three 2-Sylow subgroups of $S_3$ and find a 2-Sylow subgroup and a 3-Sylow subgroup of $S_4.$

2. The attempt at a solution

I got $|S_3| = 6 = 2\dot\ 3$ and $|S_4| = 24 = 2^3\dot\ 3.$ So $S_3$ has a a Sylow 2 subgroup of order 2 and a Sylow 3 subgroup of order 3. I am asked to find the three 2-Sylow subgroups of $S_3$ so since the 2-Sylow of $S_3$ has order 2 is it just the permutations $(1 2), (1 3), (2 3)?$

 

[pasmith]

Homework Statement

Find the three 2-Sylow subgroups of $S_3$ and find a 2-Sylow subgroup and a 3-Sylow subgroup of $S_4.$

2. The attempt at a solution

I got $|S_3| = 6 = 2\dot\ 3$ and $|S_4| = 24 = 2^3\dot\ 3.$ So $S_3$ has a a Sylow 2 subgroup of order 2 and a Sylow 3 subgroup of order 3. I am asked to find the three 2-Sylow subgroups of $S_3$ so since the 2-Sylow of $S_3$ has order 2 is it just the permutations $(1 2), (1 3), (2 3)?$

Those are the generators of the only subgroups of S_3 of order 2, so yes.