Final answer:
It is true that a group of order pq, with p and q prime, contains an element of order p. The class equation and the restrictions on the order of elements from Lagrange's Theorem help to show this is the case.
Step-by-step explanation:
You are asking whether it is true or false that a group of order pq, with p and q being prime, contains an element of order p. This is indeed true. The class equation for a finite group G is |G| = |Z(G)| + ∑ [G:C_G(g_i)] where |G| is the order of the group, |Z(G)| is the order of its center, C_G(g_i) are the centralizers of non-central elements g_i, and the sum is taken over representatives g_i of the non-central conjugacy classes. Since p and q are prime and different, the center cannot have order pq (as the group would then be abelian and all elements would be central), hence it must be 1 or p or q by Lagrange's Theorem.
If the center had order q, then the remaining p elements must form a single conjugacy class, which is not possible as the size of a conjugacy class divides the group order. Therefore, the center must have order p, which means there is an element of order p in the center of the group.