Computing the order of a group

[Benzoate]

Homework Statement

Let a be an element of a group an let |a| = 15. Compute the orders of the following elements of G

a) a^3, a^6, a^9, a^12

Homework Equations

The Attempt at a Solution

for the first part of part a, would a^3 be <a^3>=<e,a^3,a^6,a^9,a^12,a^15,a^18,a^21,a^24,a^27,a^30, a^33,a^36,a^39,a^42>

 

[NateTG]

|a|=15 means that 15 is the lowest power of a that is equal to the identity.

So, for example,
$$a^{42}=a^{15}\times a^{15} \times a^{12}=e \times e \times a^{12}=a^{12}$$
so you've got too many things in your list.

 

[matt grime]

What I want to know is why the OP stopped after a^42 in particular. I mean going beyond a^15 is clearly wrong, but why stop at a^42? Is that the 15th power of a^3? I think so, from quickly scanning the list.