. Let G = 0,1,2,3,4}. Find the order of all the elements of G under the binary operation“addition modulo 5”

The order of an element under a specific operation in a group is the smallest positive integer n such that performing the operation n times brings you back to the element you started with. In this case, the operation is "addition modulo 5".

Let's find the order of each element in G = {0,1,2,3,4}:

1. For 0: Adding 0 to itself any number of times will always give 0. So, the order of 0 is 1.

2. For 1: Adding 1 to itself modulo 5 gives the sequence 1, 2, 3, 4, 0. So, the order of 1 is 5 because it takes 5 additions to get back to 1.

3. For 2: Adding 2 to itself modulo 5 gives the sequence 2, 4, 1, 3, 0. So, the order of 2 is 5 because it takes 5 additions to get back to 2.

4. For 3: Adding 3 to itself modulo 5 gives the sequence 3, 1, 4, 2, 0. So, the order of 3 is 5 because it takes 5 additions to get back to 3.

5. For 4: Adding 4 to itself modulo 5 gives the sequence 4, 3, 2, 1, 0. So, the order of 4 is 5 because it takes 5 additions to get back to 4.

So, the order of all the elements in G under the binary operation "addition modulo 5" is 1 for 0 and 5 for 1, 2, 3, 4.