Let G be a finite group that has the properties that it contains 2 elements a and b such that:
1.
2.
3. a ≠ e
4. b ≠ e
Find O(b).
Ok, I've tried applying the process for others such as , or really any n where can be broken down into factors of and I think I'm just not seeing it. When attempting to find something such as , if I broke that down into and , then the order would matter since the group is not necessarily abelian, right? I may just be missing something or it's not catching my eye but I don't see right now how I'm going to get , for some n. Any additional "hints" or points in the right direction are appreciated
I think I get it in that if , then we have , which gives us . Thus, the order of b must be a divisor of 31, meaning it is either 1 or 31. However it can't be 1 since b is a non-identity element by the given, thus .
However, can you please explain why gives ? Thanks.