A , nite group G, in which two randomly chosen subgroups H and K commute, has been classi, ed by Iwasawa in 1941. It is possible to de, ne a probabilistic notion, which \measures the distance" of G from the groups of Iwasawa. Here we introduce the generalized subgroup commutativity degree gsd(G) for two arbitrary sublattices S(G) and T(G) of the lattice of subgroups L(G) of G. Upper and lower bounds for gsd(G) are shown and we study the behaviour of gsd(G) with respect to subgroups and quotients, showing new numerical restrictions.

In finite groups the probability that two randomly chosen elements commute or randomly ordered n-tuples of elements mutually commute has recently attracted interest by many authors. There are some classical results estimating the bounds for this kind of probability so that the knowledge of the whole structure of the group can be more accurate.The same problematic has been recently extended to certain classes of infinite compact groups in [2], obtaining restrictions on the group of the inner automorphisms. Here such restrictions are improved for a wider class of infinite compact groups.

We consider two commutativity ratios Pr (G) and f (G) in a nite group G and examine the properties of G when these ratios are ‘large’. We show that if Pr (G)>7/24, then G is metabelian and we give threshold results in the cases where G is insoluble and G’ is nilpotent. We also show that if f (G)>1/2, then f (G) =n+1/2n, for some natural number n.

R is called commuting regular ring (resp. semigroup) if for each x, yÎR there exists aÎR such that xy=yxayx. In this paper, we introduce the concept of commuting p-regular rings (resp. semigroups) and study various properties of them.