(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

In a finite group, show that the number of non-identity elements that satisfy the equation:

x^5 = e = identity element of multiplication mod n = 1

is a multiple of 4.

(Also need to show: if the stipulation that the group be finite is omitted, what can you say about the number of non-identity elements that satisfy x^5 = e ... but if I get the part of the question that's not in parenthesis, I should be able to figure it out...but I'm mentioning it just in case there's a hint in the asking of the question).

2. Relevant equations

axioms of groups: associativity, existance of inverse-elements, existance and uniqueness of identity, and we are in the (Abelian) group U(n), which is multiplication mod n. i think those are the ingrediants needed for a group.

3. The attempt at a solution

my professor suggested "look for a pattern", so i used Microsoft EXCEL to multiply mod n and find groups for U(n), for the cases n = 4, 5, 6, ... 14.

[tex]U(4) = \left\{ {1,3} \right\}[/tex]

[tex]U(5) = \left\{ {1,2,3,4} \right\}[/tex]

[tex]U(6) = \left\{ {1,5} \right\}[/tex]

[tex]U(7) = \left\{ {1,2,3,4,5,6} \right\}[/tex]

[tex]U(8) = \left\{ {1,3,5,7} \right\}[/tex]

[tex]U(9) = \left\{ {1,2,4,5,7,8} \right\}[/tex]

[tex]U(10) = \left\{ {1,3,7,9} \right\}[/tex]

[tex]U(11) = \left\{ {1,2,3,4,5,6,7,8,9,10} \right\}[/tex]

[tex]U(12) = \left\{ {1,5,7,11} \right\}[/tex]

[tex]U(13) = \left\{ {1,2,3,4,5,6,7,8,9,10,11,12} \right\}[/tex]

[tex]U(14) = \left\{ {1,3,5,9,11,13} \right\}[/tex]

Then: of these elements, find only those guys that satisfy x^5 = e = 1. The elements of U(11): 3, 4, 5, and 9, four elements, satisfy x^5 = e. Everything else, only 1, the identity element (not a non-identity element! ha ha) satisfied x^5 = e.

Too small a set of instances (one!) to find a pattern. Is there some sort of closed form expression for 3, 4, 5, and 9 "in" U(11) that raising these guys to the 5th power that I could show to be mandatorily divisible by 4 that I am missing? Any suggestions on how to approach this? Am I going down a blind alley by just looking for a pattern? :-|

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# Prove in U(n) for n finite: x^5 = e has a number of solutions that is multiple of 4

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