The commuting probability for the Sylow subgroups of finite and profinite groups

The commuting probability Pr(G, G) of a finite group G is the probability that two randomly chosen elements of a group G commute. The knowledge of Pr(G, G) gives information on the structure of G. For instance, it is known that if Pr(G, G) > 5/8 then G is abelian, if Pr(G, G) > 1/2 then G is nilpotent, if Pr(G, G) > 1/12 then G is soluble. If X and Y are subsets of a finite group G, we can define the commuting probability Pr(X, Y ) of X and Y . We will discuss how the values of commuting probability of suitable Sylow subgroups of a finite group G affect the structure of G. Some similar interesting questions can be asked also in the profinite setting.