Let $\sigma = (1 2 \dots 9) \in S_{10}$.

a) Calculate the size of the normalizer $N_{S_{10}}(<\sigma >)$.

b) Describe exactly the elements in $N_{S_{10}}(<\sigma >)$.

I am not sure how to approach this. I understand that we look for permutations that fixes $10$, and yet I can't see what to do further...

I know that $\tau \sigma \tau^{-1} =(\tau(1) \dots \tau(9))$ by definition, and also that for $\tau$ to be in $N_{S_{10}}(<\sigma >)$ than it is required that $\tau \sigma \tau^{-1} =(\tau(1) \dots \tau(9)) = \sigma^i$ for some $i$.

How can I continue from here?

  • $\begingroup$ Let $g \in N = N_{S_{10}}(\langle \sigma \rangle)$. Since $\sigma \in N$, by multiplying $g$ by a power of $\sigma$ you can assume $g(1)=1$. Now $g^i(1)=i+1$, so $g\sigma g^{-1} = g^i \Leftrightarrow g(2)=i+1$, so $g(2)=2,3,5,6,8$ or $9$. $\endgroup$ – Derek Holt Jul 25 '18 at 7:48
  • $\begingroup$ @Derek Holt It is really difficult for me to follow all of the assumptions you made... Could you please elaborate? $\endgroup$ – ChikChak Jul 28 '18 at 21:05

I'll try to continue from where you stopped...

It has to be $o(g\sigma g^{-1})=o(\sigma^i)=9$ so $gcd(i,9)=1\Rightarrow i=1,2,4,5,7,8$

The $g\in S_{10}$ with the above property (for fixed $i$) form a coset of $C_{S_{10}}(<\sigma>)=<\sigma>$ and the number of such cosets is $6$ (because we have $6$ $i$'s)

For example for the $g\in S_{10}$ with $g\sigma g^{-1}=\sigma^2$ we see that all $t\in S_{10}:t\sigma t^{-1}=\sigma^2$ are $gC(<\sigma>)=\{t: t=gu \text{ with } u\sigma u^{-1}=\sigma\}$

And since $|<\sigma >|=9$, $|N_{S_{10}}(<\sigma>)|=9\cdot 6=54$

For (b) the elements of $N_{S_{10}}(<\sigma>)$ are these of the form $g\sigma^l$ with $\sigma^l\in<\sigma> $ and $g\in S_{10}$ with the property described in the first line.

I hope it is correct!

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.