Show Z(Sn)={1} (n>=3). How can I prove this by showing it is true for the following 3 cases?

sn group of permutations.
z(sn) center of sn.

let alpha=alpha_1* alpha_2*...alpha_k represent arbitrary permutation in sn. (each alpha represents product of disjoint cycles, believe).

there 3 cases need show. in order show identity thing in center, must come beta doesn't commute alpha. here 3 cases:

case 1: there exists alpha_1 length of alpha_i @ least 3. [since disjoint cycles commute, might write @ front]

case2: alpha_i order = 2 k>=2. (a double, triple or more 2 cycle believe)

case3: alpha=(pq) [one single 2 cycle]

still trying understand question...i trying understand 3 cases , why needed. appreciate understanding question appreciate answer! thank you.

sn group of permutations. z(sn) center of sn. let alpha=alpha_1* alpha_2*...alpha_k represent arbitrary permutation in sn. (each alpha represents product of disjoint cycles, believe). there 3 cases need show. in order show identity thing in the...

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