1. INTRODUCTIONThe concept of group probability was initially proposed by Miller [1]. The commutativity degree of a finite group denoted as , is defined as the probability that any two randomly selected elements in G commute. Following Miller’s foundational work, numerous researchers have explored this topic, leading to significant advancements. For example, Erdos and Turan [2] investigated the commutativity degrees of symmetric groups, while Gustafson [3] adopted an alternative approach by demonstrating that the commutativity degree of G equals the number of conjugacy classes divided by the order of G. Through further analysis, [3] and [4] established that the maximum commutativity degree for a finite non-abelian group G is . In addition, various upper and lower bounds for diferent probabilistic measures have been derived. For instance, Pournaki and Sobhan [5] determined a lower bound for the commutativity degree, and R. Kanti & Kumar [6] examined the probability that the commutator of two group elements equals a specific element, known as the g-commutativity degree. Recent work by Moradipour et al. [7] has precisely determined the commutativity degrees for several important group families, including generalized quaternion groups, dihedral groups, semidihedral groups, and quasi-dihedral groups. Subsequent studies have further explored the commutativity degrees of groups and their extensions, as documented in [8, 9]. These investigations remain fundamentally connected to the core concept of group commutativity degree.
The coprime degree concept was developed as an extension of the group commutativity degree. In [10], Rhani et al. introduce the notion of coprime probability of a finite group. Given two elements of the group such a notion measures the chance of their orders to be coprime. The researchers initially determined probability measures for the p-groups and certain dihedral groups. Subsequently, Zulkifli and Ali [11] conducted comprehensive investigations of coprime degree in their 2019 work, focusing specifically on non-abelian metabelian groups.
In this paper, we refine the definition of coprime probability by considering unordered pairs of the group instead. The proposed formulation of coprime probability is considered more natural and reduces repetitions since, in fact, the orders of two group elements being coprime are free of their ordering. Moreover, this definition is consistent with the coprime graph of a group, which is not a directed graph and therefore does not distinguish between edges and edges . With this new definition, we then determine the number for cyclic groups, nilpotent groups, dihedral groups, and generalized quaternion groups.
2. MAIN RESULTSFirst, we provide some definitions and elementary properties in group theory. Let be a group. The cardinality of , denoted by is called the order of . For , the smallest positive integer n satisfying is called the order of denoted by . It is well known that when G is finite, divides |G|, for all . For a prime number , a finite group is said to be a p-group is a positive power of . Hence, in a -group every element has order a non-negative power of
Let be a subset of G. We say that generates , denoted with ⟩, if every element in can be written as a product of a finite number of elements in or their inverse. In this case, the set S is called a generator of . Furthermore, when , we say that G is a cyclic group. Other properties of a group can be found in, for example, [12] or [13].
Next, we have two classes of finite groups, namely dihedral groups and generalized quaternion groups [14]. For an integer , the dihedral group is a group of order 2n generated by two elements that satisfy the relations . The last relation may also be expressed as
The dihedral group is the symmetry group on regular . For an integer , the generalized quaternion group is a group of order 4n generated by two elements that satisfy the relations
We recall some structural notions that will be used throughout the paper. Two groups and are said to be isomorphic, written , if there exists a bijective homomorphism between them. In this case, the groups share identical algebraic structure and group-theoretic properties, such as element orders being preserved.
If are groups, their direct product is the group whose elements are tuples , for all with componentwise operation. The order of an element in a direct product satisfies
In particular, by the fundamental theorem of finite abelian groups, a finite cyclic group and, more generally, a finite nilpotent group can be decomposed as a direct product of its Sylow p-subgroups (see, e.g., Dummit and Foote [12] or Isaacs [13]). These structural facts play a key role in our computation of coprime degrees.
Two positive integers m, n are coprime if their greatest common divisor, denoted by , is 1. A pair of coprime numbers does not have a common prime factor. Two elements in G are coprime if their orders are coprime.
Given a finite group the set of all ordered pairs of coprime elements in is denoted as
Rhani et al. [10] defines the coprime probability of a group G as
They also established that if G is a p-group of order , then
hence
We first comment on the definition of coprime probability. Notice that if and only if . Therefore, all pairs of the same elements do not factor in determining the chance of obtaining a coprime pair of elements of G. We think it is more natural to consider instead the coprimeness of unordered pairs of elements, that is to take into account 2-subsets of G. We propose the following definition.
Definition 2.1.Let G be a finite group. The coprime degree of G is the proportion unordered pairs of elements of G whose orders are coprime, defined by
We can relate the coprime degree to the coprime probability. Let be two distinct elements of G. Whenever x and y are coprime, they appear twice, as and , in copp(G), but they appear once as an unordered pair of coprime elements in G. Also, copp(G) contains exactly one pair of the same elements, namely (e, e). Therefore, the number of unordered pairs of coprime elements in equals . The number of unordered pairs of elements in G is This establishes
In order to calculate we will utilize copp(G).
We will need the following lemma for the rest of our discussion.
Lemma 2.2.Let G be a finite group, H be any p-group of order , and If gcd , then , where
Proof. Consider the partition of copp(K) into
and
Let and , for some . Since and , it follows that copp(K) if and only if gcd if and only if . Hence, . Therefore,
Next, let and , for some Then , for some 0 , \end{document} ]]> so . It follows that copp(K) if and only if gcd and if and only if gcd and . Hence, V = ∪y∈H,y̸=e {((x, y), (x′, eH)) | (x, x′) ∈ copp(G)}. Therefore
We Conclude that
We are now ready to calculate the coprime degrees of some special groups. It is clear that we only need to determine the number of coprime pairs of each group.
2.1. Cyclic and nilpotent groups.Let n be a positive integer larger than 1. It is well-known that a cyclic group of order n is isomorphic to the additive group . Moreover, if is the prime factorization of then
Theorem 2.3.Let G be a cyclic group of order n and be the prime factorization of n. Then,
and therefore
Proof. It sufices to prove the case of . We use induction on the number m of prime factors of .
If , then G is a , and the result follows from [10].
Now, assume 1 \end{document} ]]> and the result is true for any cyclic group with prime factors. Write , where
Since and are coprime, it follows from Lemma 2.2 that | copp(G)| = . By the induction hypothesis, | copp and (5) and (6) follow immediately. □
Nilpotent groups are generalizations of cyclic groups. Theorem 8.15 in [13], a finite nilpotent group is characterized by having unique Sylow p-subgroups for each prime factor of its order, and the group is a direct product of those Sylow subgroups. Since every Sylow subgroup is a -group, it follows that a nilpotent group is a direct product of p-groups.
Using the same proof as in Theorem 2.3, we obtain the following. In particular, let G be a finite nilpotent group of order , where this is the prime factorization of n. According to [13], the group can be expressed as a direct product , where denotes the Sylow -subgroup of of order for . Theorem 2.4 can then be obtained by mathematical induction on m, the number of diferent prime factor of following steps similar to those used in the proof of Theorem 2.3.
Theorem 2.4.Let G be a nilpotent group of order n and be the prime factorization of n. Then
and therefore
Before we proceed with other classes of groups, let us observe the order of elements in a cyclic group.
Let G be a cyclic group of order n generated by an element a. Let for some integer . Then
Let s be a divisor of n such that is the highest power of s that divides n. For s not to divide , a necessary and suficient condition is that divides , hence divides k. The number of k for which is coprime with s is therefore
2.2. Dihedral groups.Let n be an integer, and let be the dihedral group of order Let be generated by a and b that satisfy $a ^ { n } = b ^ { 2 } = ( a b ) ^ { 2 } = e , { \mathrm { s e e ~ } } [ 14 ]$ . Letting
, the group are partitioned into two cosets H and Hb. Notice that all elements in Hb are of order 2.
To determine , let be the prime factorization of . Next, we partition into four subsets:
(A)
(B)
(C) , and
(D)
First notice that and since H is a cyclic group of order n, we have . Next, since , for all , we have Notice also that
To determine we need to consider two cases: 2 divides n, and 2 does not divide n.
Assume now that 2 divides n and, without loss of generality, let . Then if and only if . The number of elements in H whose order is coprime with 2 is exactly Since , it follows that
Assume now that 2 divides n, i.e., all prime factors of n are odd. Then for all and therefore
Adding the number of elements in those four subsets, we obtain | that leads to the following theorem.
Theorem 2.5.
And Therefore
Example. Consider the dihedral group of order , where The rotation subgroup ⟨a⟩ is cyclic of order while all reflections have order 2. Using Theorem 2.5 with prime factorization , where , and , we obtain
Since 2 divides n, we have
Therefore,
Hence, the coprime degree of is
2.3. General quaternion groups.Let n be an integer, and let be the generalized quaternion group of order 4n. Let be generated by a and b that satisfy ， see [14]. From these relations, we will deduce some order properties of elements in
First, notice that and cannot be the identity, else . For nonnegative integer i we have . In particular, taking leads to hence . Since neither nor , we have . We then have . Since and , the order of a is at least n, but since it must be
As in the case of dihedral groups, we partition into two cosets and Hb. Next, we partition copp into four subsets with respect to membership of elements in the pair in H or Hb. Let be the prime factorization of 2n with . Proceeding as in the case of dihedral groups, i.e. by partioning the set ) into 4 subsets, we obtain the following theorem.
Theorem 2.6.
and therefore