Suppose that a is a group element and a6 e What are the pos

Suppose that a is a group element and a^6 = e. What are the possibilities for |a|? Provide justification for your answer.

Solution

As it is mentioned in the question that a⁶ = e then we have a such that a is 1, 2, 3, 4, 5, 6 as whatever be the value of IaI be , it can never be greater than 6.

But let us see that all the aobe possibility satisfy the given condition or not.

We can see that if a⁴ = e........(1) then

a⁴ = a².e = a^{2 [as e is identity when multiplied the quantity remains same]}

Therefore a² = a². a⁶ = a⁸ = (a⁴)² = (e)²  = e [ by (1) and a⁶ = e is given to us]

Hence IaI = 4 is not possible.

Let us suppose that a⁵ = e then

a⁵ = e = a³. e² = a³. (a⁶)² = a¹⁵ = (a⁵)³ = (e³)= e

Hence IaI = 5 is also not possible

so possible values of a = 1, 2, 3, 6

Note: One more way to check our answer is that all the values of a we got are divisors of a while

a = 5, 4 were not divisors of a. So our answer is correct.