Please can someone help me with the following problems:
a and b are distinct primes such that a<b, and H is a finite group such that
l H l = ab
1) How can I use Sylows theorem to show that H has a normal subgroup K with K being isomorphic to Cp (C is the cyclic group).
2) Describe explicitly all homomorphisms h: C5 --> Aut(C7), and from this, describe all groups of order 35. How many such subgroups are there?
3) Same as 2) but this time h: C3 --> Aut(C13), and from this, describe all groups of order 39. Again, how many such subgroups are there?