Let G = 1,2,3,4,5,6}. Find the order of all the elements of G under the binary operation“multiplioation modulo 7”

To find the order of all the elements of G under the binary operation "multiplication modulo 7", we need to find the smallest positive integer n such that a^n ≡ 1 (mod 7) for each a in G.

1. For a = 1, 1^n ≡ 1 (mod 7) for all n, so the order of 1 is 1.
2. For a = 2, we find that 2^3 ≡ 1 (mod 7), so the order of 2 is 3.
3. For a = 3, we find that 3^6 ≡ 1 (mod 7), so the order of 3 is 6.
4. For a = 4, we find that 4^3 ≡ 1 (mod 7), so the order of 4 is 3.
5. For a = 5, we find that 5^6 ≡ 1 (mod 7), so the order of 5 is 6.
6. For a = 6, 6^n ≡ 1 (mod 7) for all n, so the order of 6 is 1.

So, the orders of the elements 1,2,3,4,5,6 under the operation "multiplication modulo 7" are 1, 3, 6, 3, 6, 1 respectively.