I've got problem with permutation group multiplication.
Here is an example:
Determine the permutation $\alpha = S_9 $ is that $ \alpha*\omega * \alpha^{-1} = \gamma$ . How much of those permutations we have?
$ \omega= (13624)(587)(9) $ and $\gamma = (15862)(394)(7)$
$ \alpha*\omega * \alpha^{-1} = \gamma$
Is here any trick to do this? I will be very thankful for every help.
Yes, there is a trick, or more specifically a theorem. It says that for $(i_1,...,i_k) \in S_n$, $$\alpha\cdot (i_1,...,i_k) \cdot \alpha^{-1}=(\alpha(i_1),...,\alpha(i_k)).$$ In particular, $$\alpha \omega \alpha^{-1}=\alpha (13624)\alpha^{-1}\alpha (587)\alpha^{-1}\alpha (9)\alpha^{-1}=(15862)(394)(7).$$ So you're looking for $\alpha \in S_n$ such that $\alpha(1)=1,\alpha(3)=5,\alpha(6)=8$, etc...
