What is the difference between a proper subgroup and a subgroup?

1. Consider the group U(5).(a) What is |U(5)|?(b) For each a ∈ U(5), find |a|, < a >, and | < a > |   2. Consider the group Z6.(a) What is Z6?(b) For each a ∈ Z6, find |a|, < a >, and | < a > |.

Let G be a group, let H≤G be a subgroup, and let NG be a normal subgroup. (i) Show that HnN is a normal subgroup of H. (ii) Suppose now that H is a normal subgroup of G. Show that HnN is a normal subgroup of G.

Prove that a normal subgroup must be a union of conjugacy classes.