Group Theory: Modular Arithmetic and Linear Functions

In this problem, we'll explore group theory. First, we'll work with a finite set of integers under modular addition, and then we'll prove that a specific set of linear functions forms a group under composition.

Part one gives us a set S containing an unknown element 'a', along with 2, 4, and 6. This set forms a group under addition modulo 8. For any group, the identity element must be present in the set.

In the group of integers under addition modulo 8, the identity element is zero. Therefore, 'a' must be equal to 0 for the set to satisfy the identity axiom of group theory.

Now, let's construct the Cayley table for this set under addition modulo 8. We perform addition and take the remainder when divided by 8.

We arrange our elements 0, 2, 4, and 6 in rows and columns. Adding 0 to any element keeps it the same. Adding 2 plus 2 gives 4, 2 plus 4 gives 6, and 2 plus 6 is 8, which is 0 in modulo 8.

Observe that every row and column contains each element exactly once, and all results are within the set, confirming it is indeed a group.

Now let's move to the second part. We want to show that functions of the form f of x equals p x plus q form a group under composition. To prove this, we must check four axioms: closure, associativity, identity, and inverse.

First, closure. Let's take two functions, f and g. Their composition involves plugging g into f.