Question

In Abstract Algebra there is a structure referred to as a field. A field is a set along with two binary operations, we will call them addition and multiplication. The set along with the addition operation forms a group and the non-zero elements along with multiplication form a group. a. Show Z 5 form a group under the ⊕ modular addition operation.

Answer 1

To show that Z5 forms a group under the modular addition operation (⊕), we need to demonstrate closure, associativity, identity, and inverse.

Step-by-step explanation:

In Abstract Algebra, a field is a set along with two binary operations: addition and multiplication. To show that Z5 forms a group under the modular addition operation (⊕), we need to demonstrate the following:

1. Closure: For any two elements a, b in Z5, a ⊕ b is also in Z5.
2. Associativity: For any three elements a, b, and c in Z5, (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c).
3. Identity: There exists an element 0 in Z5 such that for any element a in Z5, 0 ⊕ a = a ⊕ 0 = a.
4. Inverse: For any element a in Z5, there exists an element b in Z5 such that a ⊕ b = b ⊕ a = 0.