Exercise
Consider the permutation group G consisting of the following 8 elements:
(ab)2 = (1,3)(2,4), bab = (2,4), aba = (1,4)(2,3).
What is its center?
{e, a}
{e, (ab)2}
{e, aba}
{e, bab}
No.
For instance,
bab-1 = bab 
a.
a.
Correct. Verifying that a and b both centralize (ab)2,
and realizing that the centralizer is a group, we find that the subgroup of G
generated by a and b centralizes this group. But that subgroup is G.
Some computations show that each other element is moved under conjugation
by a suitable element to a different element, and hence does not belong to Z(G).
No.
For instance,
a(aba)a-1 = b
aba.
aba.
No.
For instance,
b(bab)b-1 = a
bab.
bab.