In this paper we introduce and study the concept of cyclic subgroup commutativity degree of a finite group G. This quantity measures the probability of two random cyclic subgroups of G commuting. Explicit formulas are obtained for some particular classes of groups. A criterion for a finite group to be an Iwasawa group is also presented.
In this paper we study probabilistic aspects such as subgroup commutativity degree and cyclic subgroup commutativity degree of the (generalized) dicyclic groups. We find explicit formulas for these concepts and we provide another example of a class of groups whose (cyclic) subgroup commutativity degree vanishes asymptotically.
Let C(G) be the poset of cyclic subgroups of a finite group G and let P be the class of p-groups of order p n (n ≥ 3). Consider the function α : P −→ (0, 1] given by α(G) = |C(G)| |G| . In this paper, we determine the second minimum value of α, as well as the corresponding minimum points. Further, since the problem of finding the second maximum value of α was completely solved for p = 2, we focus on the case of odd primes and we outline a result in this regard.MSC (2010): Primary 20D60; Secondary 20D15, 20D25.
In this paper we show that there is an infinite number of finite groups with two relative subgroup commutativity degrees. Also, we indicate a sufficient condition such that a finite group has at least three relative subgroup commutativity degrees and we prove that D6 is the only finite dihedral group with two relative commutativity degrees. Finally, we study the density of the set containing all subgroup commutativity degrees of finite groups.
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