How many elements in $S_{8}$ are conjugate with $(12)(345)$?

Question

How many elements in $S_{8}$ are conjugate with $(12)(345)$?

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My reasoning is as follows:

Two elements in $S_n$ are conjugate if and only if they have the same cycle type, so we need to count the number of elements of the type:

$(x_1 x_2)(x_3 x_4 x_5)$

There are $\frac{8\cdot7}{2}\cdot \frac{6 \cdot 5 \cdot 4}{3}=1120$ elements of this type. So $1120$ elements are conjugate with $(12)(345)$.

Is this correct? 5

Answer 1

Yes, your reasoning and answer are correct.

Answer 2

Yes, your solution is correct. The number of ways to choose the three fixed points is ${8 \choose 3}$, the number of ways to choose the transposition from the remaining $5$ points is ${5 \choose 2}$, and the number of ways to form a $3$-cycle from the remaining $3$ points is $(3-1)!=2$. Thus, the number in question is ${8 \choose 3} \cdot {5 \choose 2} \cdot 2 = 1120$.