Exercise
Let g and h be two permutations in Sn.
A: g and h
have the same cycle structure if and only if one is a conjugate of the other.
B: g and h commute if and only if they have the same cycle structure.
They are both correct.
They are both incorrect.
A is incorrect.
B is incorrect.
Whether or not two permutations commute can not be decided on the basis
of their cycle structure (unless one of them is the identity).
How about A?
Review what the effect is of a conjugation.
Yes, whether or not two permutations commute can not be decided on the basis
of their cycle structure (unless one of them is the identity).