Number of elements in S13 conjugate to both (12)(34) and (123)(45) then is the answer 0? Because (12)(34) and (123)(45) have not got the same cycle structure.

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deginasiba · Accepted Answer

Step 1 That is correct. For this example, also notice that the two elements you mention have different orders. (12)(34) has order 2 while (123)(45) has order 6. Given a group G suppose that a,b∈G are conjugate. Then there exists c∈G such that cac−1=b If a has order n, that is if an=e, then bn=(cac−1)=cac−1cac−1⋯cac−1 =canc−1=cc−1 =e So that ord(b)≤ord(a). By summetry, we also have that ord(a)≤ord(b) and hence ord(a)=ord(b)

Number of elements in S_{13} conjugate to both (12)(34) and

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