Hey all. I am just trying to figure out the following. It's the last thing I need to revise before my exam. Any help would be fantastic. Thank you! James

Take a group of order 117. This has prime factors 3 and 13. Prove that every group with order 117 has an index 3 cyclic subgroup (normal) and a centre which is not the identity

Take a group of order 118. This has prime factors 2 and 59. Prove that every group with order 118 has an index 2 cyclic subgroup (normal) and a centre which is not the identity

I think that these make use of the Sylow Theorems at some point but I am not sure exactly how to use them

Thank you