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Let $n \geq 3$. Then there exists an element $f \in S_n$ such that $f \neq g^3$ for any element $g \in S_n$, where $S_n$ denotes the symmetric group on $n$ letters. How to establish whether this assertion is true or false?

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2 Answers 2

Hints:

  • For any $\,n\ge 3\;$ there are elements of order $\;3\;$ in $\;S_3\;$

  • The map $\;\phi_3:S_n\to S_n\;,\;\;\phi_3(\sigma):=\sigma^3\;,\;\;n\ge3\;,\;$ cannot be injective...

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Thanks, but I'm afraid I've not picked the hint. I guess we require the map to be not surjective. –  Saaqib Mahmuud Sep 15 '13 at 7:17
1  
Well, if $\;X\;$ is a finite set, then a map $\;f:X\to X\;$ is injective iff it is surjective... –  DonAntonio Sep 15 '13 at 9:22

Prove that $(123)\neq \sigma^3, \ \forall \sigma\in S_n$ where $(123)$ stands for the 3-circle $1\mapsto2, \ 2\mapsto3, \ 3\mapsto1$. To prove it use that if $(123)= \sigma^3$, then the order of $\sigma$ must be $9$ and therefore $\sigma$ must be a (product of) $9$ circle(s). Then derive a contradiction.

In general for $n\geq m, \ (123\cdots m)\neq\sigma^m, \ \forall \sigma\in S_n$.

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Lol, so right! Here's to hoping you join the 10k club soon. –  Jyrki Lahtonen Sep 14 '13 at 13:49
    
Welcome to the 10k club :) –  Clive Newstead Sep 15 '13 at 3:10
    
Thanks, but I didn't get it. –  Saaqib Mahmuud Sep 15 '13 at 7:19

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