C8. (12 points) In this question, we will explore how the Linear Congruence Theorem has applications in other areas of mathematics. To do this, we will define a term from group theory: Definition C8. A group is a set G equipped with an operation ., satisfying the follow- ing properties: . There exists an identity 1 in G such that for any g E G, we have 1 . g = g . 1 = 9; . For any g E G, there exists an inverse go such that g . g ) = g ] . g = 1. Furthermore, we can say that S is a subgroup of G if it has the following properties: . S is a subset of G, meaning every element of S is also an element of G; . S is itself a group, with the same operation . that G has, and the same identity 1; . S is closed under .: for any a, be S, we have a . be S. Let p be a prime and a E Zp such that a # 0. Consider the group G = {a", al, ..., ap-?} equipped with the usual multiplication from Zp. You can take it for granted that a" = 1 is the identity of G. Use the Linear Congruence Theorem to prove the following: If d | (p - 1), then G contains a subgroup with d elements. (Hint: how many incon- gruent solutions are there to dr = 0 (mod p - 1)? Use these solutions to determine which powers of a should be in the subgroup. You must then prove that your pro- posed subgroup meets all the requirements to be a subgroup.)