In $S_7$, describe the conjugates of $(1 2)(3 4 5)$
What I have done
I'm not quite sure what the textbook refers to by "describe", in any case, I calculated (?) how many of them there are. it is right? If not, what should I actually do?
Let $\sigma=(1 2)(3 4 5)$ and $\tau$ any permutatión in $S_7$, then $\tau \sigma \tau^{-1}$ has the same cycle structure of $\sigma$. Since $\tau \sigma \tau^{-1}= (\tau(1) \tau(2))(\tau(3) \tau(4) \tau(5)) $, for $\tau(1)$ we have $7$ possibilities and $6$ possibilities for $\tau(2)$. Since $(\tau(1)\tau(2))=(\tau(2)\tau(1))$, we have $(6\cdot7)/2=21$ $2$-cycles. Similarly, we have $(5\cdot 4\cdot 3)/3=20$ $3$-cycles. So in total, there are $21\cdot 20=420$ conjugates of $(1 2)(3 4 5)$.