Permutation on Sets involving relations and composition of functions (hard math)?

T

TheKid

New member
given non-empty set A and P(A) is a set of all permutations of A. For example, P(A) = { f is an element of set A going to set A such that f is bijective }. Relation ~ on P(A) by setting
f ~ g iff there exists h is an element of P(A) such that h o f o h^(-1) = g
for f, g is an element of P(A)

o is notation for composition of functions
^(-1) means inverse

So if A = {0, 1} then how many elements does P(A) have? I'm thinking 2. What about P(A) / ~ ? I'm still thinking 2.
What if A = {0,1,2} then how many elements does P(A) have? I'm thinking 3. Again, what about P(A) / ~ ? I'm thinking 2.
Help?
 
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