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

Susan Nall

Susan Nall

Answered question

2022-01-14

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.

Answer & Explanation

deginasiba

deginasiba

Beginner2022-01-15Added 31 answers

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,bG are conjugate. Then there exists cG such that cac1=b
If a has order n, that is if an=e, then
bn=(cac1)=cac1cac1cac1
=canc1=cc1
=e
So that ord(b)ord(a). By summetry, we also have that ord(a)ord(b) and hence ord(a)=ord(b)

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