Assignment:

1.  What are the 2-sylow subgroups of S4.

ans:  They are the symmetries of the square:

{(1234), (1432), (13), (24), (12)(34), (13)(24), (14)(23), I}
{(1243), (1342), (14), (23), (12)(34), (13)(24), (14)(23), I} 
{(1324), (1423), (12), (34), (12)(34), (13)(24), (14)(23), I}

2.  What are the 3-sylow subgroups of S4.
ans: They are the subgroups generated by 3-cycles.

{(123),(132),I}
{(124),(142),I}
{(134),(143),I}
{(234),(243),I}

3.  Show that every group of order 20 has a normal subgroup.
Divisors of 20     1    2    4     5   10   20
2-Sylow            x               x
5-Sylow            x

The 5-Sylow subgroup with 5 elements is normal

4.  Show that every group of order 75 has a normal subgroups.
Divisors of 75     1  3   5   15  25  75
3-Sylow            x               x
5-Sylow            x                   

The 5-sylow subgroup of order 25 is always normal.