- #1
Mr Davis 97
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Homework Statement
Prove that if ##1 \leq d \leq n##, then ##S_n## contains elements of order d.
Homework Equations
The Attempt at a Solution
Here is my idea. The order of the identity permutation is 1. Written in cycle notation, the order of (1,2) is 2, the order of (1,2,3) is 3, the order of (1,2,3,4) is 4, and in general the order of (1,2,3,4,...,n-1,n) is n. It would seem that this shows that if ##1 \leq d \leq n##, then ##S_n## contains elements of order d.
However, I am not sure how rigorous this is. Do I need to make an induction argument that in general (1,2,3,4,...,n-1,n) has order n, or is it fine as it is?