We know in $S_4$, the conjugacy classes are $$(1) \quad (1 2) \quad (123) \quad (1234) \quad (12)(34)$$ The thing I don't understanding the how to calculate the size. For $(1)$, it is obviously, having size 1. But for others, for example, why size of $(1234)$ is $3 \times 2=6$, size of $(123)$ is $4 \times 2=8$
I would like to know how to count the size effectively.
Thank you!
$\endgroup$1 Answer
$\begingroup$Note if an element $\theta$ is conjugate to $(1234)$ iff it has the same cycle structure. Now $(1234)$ is a $4$-cycle so all $4$-cycles will be conjugate to $(1234)$ and the number of $4$-cycles is $(4-1)!=3!=6$. And if you to calculate the conjugacy class of $(123)$ then you have to count the number of $3$-cycles in $S_{4}$. Number of $3$-cycles can be counted in the following manner. The number of ways to select $3$ elements is $\binom{4}{3}$ and each element can be permuted in $2!$ ways among themselves. Hence you have $8$ $3$-cycles in $S_{4}$.
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