On the commuting probability for subgroups of a finite group

Let be a subgroup of a finite group. The probability that an element of commutes with an element of is denoted by. Assume that for some fixed 0. We show that there is a normal subgroup and a subgroup such that the indices and and the order of the commutator subgroup are -bounded. This extends the well-known theorem, due to P. M. Neumann, that covers the case where. We deduce a number of corollaries of this result. A typical application is that if is the generalized Fitting subgroup then has a class-2-nilpotent normal subgroup such that both the index and the order of the commutator subgroup are -bounded. In the same spirit we consider the cases where is a term of the lower central series of, or a Sylow subgroup, etc.