Cyclic subgroup commutativity degrees of finite groups

Abstract

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.

Probabilistic aspects of ZM-groups

“…Section 4 covers the main results of the paper. More exactly, we find an explicit formula that allows us to compute the (cyclic) subgroup commutativity degree of ZM-groups, we show that our results generalize the ones obtained for dihedral groups in [16,19], and we indicate a class of groups whose (cyclic) subgroup commutativity degree vanishes asymptotically. We end the paper by suggesting some open problems in the last section.…”

“…Using Lemma 4.1., the quantities f m1,n1 can be computed for each divisors m 1 of m and n 1 of n. Consequently, the subgroup commutativity degree of a ZM-group is given by the following result. In particular, we have sd(ZM (9, 4, 8)) = 13 19 .…”

Relative cyclic subgroup commutativity degrees of finite groups

“…Hence, |Im f 1 | = 1 ⇐⇒ csd(G) = 1 ⇐⇒ G is an Iwasawa group (i.e. a nilpotent modular group), the second equivalence being indicated in [20]. Our next result is a criterion which indicates a sufficient condition such that a finite group G has at least 3 relative cyclic subgroup commutativity degrees.…”

“…Proof. Explicit formulas for computing the cyclic subgroup commutativity degree as well as the structure of the poset of cyclic subgroups for each class of finite groups in which we are interested in are provided by [20]. Also, information concerning the number of normal subgroups contained in such groups can be found in [19].…”

“…Also, in [18], the author suggests (see Problem 3.6) to study the restriction of the function sd : L(G) × L(G) −→ [0, 1] to L 1 (G) × L 1 (G), where L 1 (G) denotes the poset of cyclic subgroups of G. Our aim is to provide some answers to this open problem. The idea of working only with cyclic subgroups instead of taking into account all subgroups of a finite group G was applied in [20], where the cyclic subgroup commutativity degree of G, defined as csd(G) = 1 |L 1 (G)| 2 |{(H, K) ∈ L 1 (G) 2 | HK = KH}|, was studied. This concept led to some fruitful results and this is an additional argument to introduce and study the restriction of sd : L(G) × L(G) −→ [0, 1] to L 1 (G) × L 1 (G).…”