I'm reviewing for a midterm, and one of the questions is:
"What is the smallest such n so contains an element of order 30? Give reasons."
I don't know how to solve this. Clearly the order of the group must be at least 30, so n is greater than 4. Also 30 must divide the order of the group, by Lagrange's Theorem, but it does for every n greater than 4.
Thanks for any help!
Pardon? is not cyclic! It is two-generated!
What you are supposed to figure out in this question is what the order of a permutation is. You know that a 2-cycle has order 2, a 3-cycle has order 3, etc. but you need to figure out what happens when you concatenate two or more cycles.
So, what is the order of ? Why?
Similarly, what is the order of , and why?
What about ?
Your solution will involve splitting 30 down into prime factors.
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(I would like to point out at this stage that the element in with highest order is an element if the form ).
Exactly. The thing you should notice most is that it does not matter what numbers are in the disjoint cycle, it is what the disjoint cycle looks like. The same thing, curiously, happens for conjugacy classes,
Two permutations are conjugate in if and only if they have the same disjoint cycle shape.
For instance, , and are all conjugate in where .