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#31
Nov23-11, 05:29 PM
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Quote Quote by AdrianZ View Post
One more question, Is it always possible to solve an equation like axn=b in Sn? When it's possible?
Quote Quote by AdrianZ View Post
Would you explain more please?
Explained with orders:

In S4, x^4 is either identity or a 3-cycle (with order 3).
If a and b differ in order, but not by 3, there is no solution.


Explained with even and odd permutions:
(Do you know what even and odd permutations are?)

In S4, x^4 is always an even permutation.
If a is odd and b is even, then there is no solution.



Quote Quote by AdrianZ View Post
We found out that there are 1 one-cycle, 6 different 2-cycles, 8 different 3-cycles and 6 different 4-cycles in S4. but if we add 1+6+8+6 it'd be equal to 21, not 24. How so?
Yes, you're missing 3 of them.
Did you already have them in your original solution?

As a challenge (when you find them), how should you count how many there are?