{0, 1, 2, 3, 4, 5} is a group under addition modulo 6, then the order of 1 is ___________.Correct answer is '6'. Can you explain this answer?

Question

Answer

Order of an element in a group under addition modulo 6

In the given group {0, 1, 2, 3, 4, 5} under addition modulo 6, we need to find the order of the element 1.

Definition: The order of an element in a group is the smallest positive integer 'n' such that when the element is repeatedly added to itself 'n' times, it equals the identity element of the group.

Identity element: In the given group, the identity element is 0. Adding 0 to any element in the group does not change the value of that element.

Order of 1: We need to find the smallest positive integer 'n' such that 1 + 1 + 1 + ... + 1 (n times) = 0.

Calculating the order of 1:
1 + 1 = 2
1 + 1 + 1 = 3
1 + 1 + 1 + 1 = 4
1 + 1 + 1 + 1 + 1 = 5
1 + 1 + 1 + 1 + 1 + 1 = 0

From the above calculations, we can observe that when we add 1 to itself 6 times, we get the identity element 0. Therefore, the order of 1 in the given group is 6.

Explanation: The order of an element in a group gives us the number of times we need to repeat the element under the group operation to get the identity element. In this case, adding 1 to itself 6 times results in the identity element 0. This means that the element 1 has an order of 6 in the given group under addition modulo 6.

Conclusion: The order of 1 in the group {0, 1, 2, 3, 4, 5} under addition modulo 6 is 6.

Answer

N(a) is order in addition module so 6(1)= 6

Answer

(1)⁶ = 1 + 1 + 1 + 1 + 1 +1 = 6 (mod 6) = 0
∴ Older of 1 is 6.

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