Find all the generators of the oyolio group G = 0,1,2,3,4} under the binary operation “additionmodulo 5”

To find the generators of the cyclic group G = {0,1,2,3,4} under the binary operation "addition modulo 5", we need to find all elements 'a' in G such that every other element in G can be written as a power of 'a'.

Step 1: Choose an element from the group. Let's start with 1.

Step 2: Generate all elements of the group by adding 1 to itself modulo 5.

1^1 = 1 (mod 5) 1^2 = 2 (mod 5) 1^3 = 3 (mod 5) 1^4 = 4 (mod 5) 1^5 = 0 (mod 5)

We see that we can generate all elements of the group G with 1, so 1 is a generator of G.

Step 3: Repeat the process for the remaining elements of the group.

For 2: 2^1 = 2 (mod 5) 2^2 = 4 (mod 5) 2^3 = 1 (mod 5) 2^4 = 3 (mod 5) 2^5 = 0 (mod 5)

So, 2 is also a generator of G.

For 3: 3^1 = 3 (mod 5) 3^2 = 1 (mod 5) 3^3 = 4 (mod 5) 3^4 = 2 (mod 5) 3^5 = 0 (mod 5)

So, 3 is also a generator of G.

For 4: 4^1 = 4 (mod 5) 4^2 = 1 (mod 5) 4^3 = 4 (mod 5) 4^4 = 1 (mod 5) 4^5 = 0 (mod 5)

So, 4 is not a generator of G because it cannot generate all elements of G.

For 0: 0^1 = 0 (mod 5) 0^2 = 0 (mod 5) 0^3 = 0 (mod 5) 0^4 = 0 (mod 5) 0^5 = 0 (mod 5)

So, 0 is not a generator of G because it cannot generate all elements of G.

Therefore, the generators of the group G = {0,1,2,3,4} under the binary operation "addition modulo 5" are 1, 2, and 3.