Suppose a group G acts on itself by conjugation such that g.a = gag-1. What is the stabilizer of an element a ∈ G?

Let G be a finite group of order n. Cayley's theorem states that G is isomorphic to a subgroup of the symmetric group Sk. What is the smallest general value of k that guarantees this embedding for any group of order n?

The order of the element (5, 3) in the direct product group ℤ30 ⊕ ℤ12 is __________.