Permutability degrees of finite groups

Given a finite group $G$, we introduce the \textit{permutability degree} of $G$, as $$pd(G)=\frac{1}{|G| \ |\mathcal{L}(G)|} {\underset{X \in \mathcal{L}(G)}\sum}|P_G(X)|,$$ where $\mathcal{L}(G)$ is the subgroup lattice of $G$ and $P_G(X)$ the permutizer of the subgroup $X$ in $G$, that is, the subgroup generated by all cyclic subgroups of $G$ that permute with $X\in \mathcal{L}(G)$. The number $pd(G)$ allows us to find some structural restrictions on $G$. Successively, we investigate the relations between $pd(G)$, the probability of commuting subgroups $sd(G)$ of $G$ and the probability of commuting elements $d(G)$ of $G$. Proving some inequalities between $pd(G)$, $sd(G)$ and $d(G)$, we correlate these notions.


Introduction
All the groups of the present paper are supposed to be finite. Given a group G and its subgroup lattice L(G), the subgroup commutativity degree of G have been largely studied in the last years. Fundamental properties and interesting generalizations of sd(G) can be found in [11,12,13,22,27,28,29], and for d(G) in [1,2,6,8,9,10,14,17,18,19,20,24]. To study these notions, various perspectives have been considered in literature, because both measure theory and combinatorial techniques may be applied in order to get restrictions on the structure of a group. The present paper investigates a similar concept, the permutability degree of G pd(G) = 1 |G| |L(G)| and its connections with sd(G) and d(G). In the previous formula, the permutizer P G (X) of a subgroup X of G is defined to be the subgroup generated by all cyclic subgroups of G that permute with X, that is, P G (X) = g ∈ G | g X = X g . This means that X ∈ L(P G (X)) and X = P G (X) if and only if X g = g X for some g ∈ G − X.
We concentrate on permutizers because several classifications are available in literature on this topic. Recall that a group G such that X = P G (X) for every proper subgroup X of G is said to satisfy the permutizer condition P, or briefly P-group. Therefore the permutizer condition generalizes the well-known normalizer condition (see [26]) and gives information on how the group is near to be supersolvable. The study of permutizers is not new and it is based on a series of fundamental contributions [3,21,23,30] in the last 20 years. From [3,Corollary 2], we know that for groups of odd order the permutizer condition is equivalent of being supersolvable and actually, a complete classification of P-groups can be found in [3]. Now we may define the subgroup and correlate it with other subgroups of G. For instance, it is easy to check that the norm N (G) of G (see [26] for the properties of N (G)) satisfies the following relation This relation emphasizes how P (G) is connected with other subgroups, widely investigated in literature, such as intersections of normalizers or of centralizers. Note that for any X ∈ L(P (G)) one has P G (X) = G. The subgroup P (G) is also important because it allows us to "manipulate" the expression of pd(G), for getting some analogies with [22,27] and of C G (x) = {y ∈ G | xy = yx} in [2,8]. This manipulation of the expression of pd(G) will allow us to detect whether P (G) is cyclic or not by looking only at the size of pd(G).

Basic properties and terminology
Some of the following observations will be useful later on.
Remark 2.1. Since P G (X) is a subgroup of G, for all X ∈ L(G), and it always contains the trivial subgroup, then |P G (X)| ≤ |G| and also 0 < Remark 2.2. A group G has pd(G) = 1 if and only if the sum of all |P G (X)| for X ∈ L(G) is equal to |G||L(G)|. By default, a quasihamiltonian group G, that is, a group in which every subgroup is permutable, has pd(G) = 1. A classification of quasihamiltonian groups can be found in [26,Theorems 2.4.11 and 2.4.16] and, roughly speaking, these groups are direct products of abelian groups by a copy of the quaternion group of order 8. In particular, abelian groups have permutability degree equal to 1.
Another case in which the permutability degree reaches 1 is the following. Remark 2.3. A P-group G in which all proper subgroups are maximal has pd(G) = 1. In such a case for all proper subgroups X of G one has X ⊂ P G (X) = G and so pd(G) = 1. One might be tempted to think that all P-groups have permutability degree equal to 1, but Example 3.2 below shows this is false, and then the additional condition "in which all proper subgroups are maximal" cannot be omitted. Now we rewrite the original expression of permutability degree in the following more useful form. Since L(P (G)) is a sublattice of L(G), it turns out that Note also that a cyclic group C (or better a quasihamiltonian group Q) has sd(C) = pd(C) = d(C) = 1 (or better sd(Q) = pd(Q) = 1). Therefore, relations between sd(G) and pd(G) are meaningful when G is noncylic and nonquasihamiltonian (see Remark 2.2).
For the sake of completeness, we recall some results of Beidleman and Heineken in [4]. The quasicenter Q(G) of G is the subgroup of G generated by all elements g ∈ G such that g K = K g , where K is an arbitrary subgroup of G. The subgroup Q(G) was introduced by Mukherjee and studied by several authors in the last years (see [4,5,25]), who investigated chains of quasicenters and relations with supersolvable groups. On the other hand, the hyperquasicenter of G, denoted by Q ∞ (G), is the largest term of the chain Recall that a normal subgroup N of G is said to be hypercyclically embedded in G if it contains a G-invariant series whose factors are cyclic. It is easy to see that G contains a unique largest hypercyclically embedded subgroup, which we denote Σ(G). More precisely, [4,Theorem 1] shows that Σ(G) = Q ∞ (G) is true for any group G. Some interesting connections hold between P-groups, P (G) and Q ∞ (G). For instance, [3, (3.1), p. 697] shows that a group G is a P-group if and only if G/Σ(G) is a P-group. As a first consequence, a group G is a P-group if and only if G/Q ∞ (G) is a P-group. As a second consequence, Z(G) ⊆ Q(G) ⊆ P (G) is true for any group G. Furthermore, Q ∞ (G) = P (G) if and only if P (G) = Σ(G).

Examples
Now we specify some of the previous notions for the symmetric group S 3 on 3 objects. This will help us to visualize analogies and differences between permutability degrees, subgroup commutativity degrees and commutativity degrees.
It is interesting to note that an example as easy as this has a lot of properties in our perspective of study. The group S 3 is supersovable by looking at the series {1} ⊳ A 3 ⊳ S 3 , but is not quasihamiltonian, due to HK = KH. At the same time, S 3 does not satisfy the normalizer condition, since it is not nilpotent. Moreover, A direct calculation shows that , agreeing with the computations in [27, p.2510] and [8,9].
Another easy (but interesting) example is the following.
The normal subgroups are supersolvable and it is also a P-group, but nevertheless its permutability degree is different from 1, because More precisely, The value of d(D 8 ) can be found in [8] and that of sd(D 8 ) in [27]. This example shows that there exist P-groups with permutability degree different from 1. Note that D 8 satisfies 8 = |D 8 | < |L(D 8 )| = 10 but Examples 3.1 and 3.2 illustrate a series of problems for the computation of the permutability degree, arising from the nature of the subgroup lattice of the groups under consideration. We will come back to this point later on.

General properties of the permutability degree
We note that [8,Theorems 2.5,3.3] shows that the commutativity degree is monotone. This is a well-known property, which is due to the fact that we are dealing with a positive monotone measure of probability. Similar situations can be found for sd(G) in [27, Proposition 2.4, Corollaries 2.5, 2.6, 2.7, Theorems 3.1.1, 3.1.5] and in [9,14,22]. For pd(G) we have something similar.  A classic splitting result for the product probability of two independent events is described by the following corollary. The proof may be generalized to finitely many factors, whose orders are pairwise coprime. Proof. Given three groups A, B and C such that A × B ⊆ C, we know that . This holds similarly for the permutizers and it is easy to see that P C (A × B) = P C (A) × P C (B). Now this fact and the assumption gcd(|G|, |H|) = 1 allow us to conclude that 1 The underlying problem we deal with is the order of the subgroup lattices, which is hard to predict in general. If we concentrate on some groups arising from finite geometries, then the situation is more clear (dihedral groups, semidihedral groups and generalized quaternion groups were studied in [2,8,11,22,27,28,29] from a similar perspective). Having in mind Examples 3.1 and 3.2, we observe from [26, pp. 26-29] and [22,27] that the dihedral group of symmetries of a regular polygon with n ≥ 1 edges has order 2n and splits in the semidirect product of a cyclic group y ≃ C n of order n by a cyclic group x ≃ C 2 of order 2 acting by inversion on C n . In particular, S 3 ≃ D 6 for n = 3 and one can note that the Hasse diagram of L(D 6 ) = L(S 3 ) forms a diamond in which there are only 4 atomic elements (see [26] for this terminology) in between {1} and D 6 and their number can be easily computed. The following Fig.III.3 summarizes the information of Example 3.1 in a more general situation. D 2p  The reader has probably noted that we used the formula for the sum of a geometric series in the previous expression for σ(p m ). Then we may conclude that The next result shows an upper bound for pd(G), when |L(G)| is of type (4.4). Proof. If P G (X) = G for some X ∈ L(G), then |G| = |P G (X)| = |P (G)| and G would be cyclic, contradicting our assumption. Without loss of generality we may assume P G (X) = G for all X ∈ L(G). The minimality of p implies that |P G (X)| ≤ |G| p for all X ∈ L(G). Of course, |L(P (G))| = 2 and (2.1) becomes This gives, as claimed.

Some theorems of structure
The present section is devoted to prove restrictions on P (G), arising from exact bounds for pd(G), when G is an arbitrary group. The evidences of Examples 3.1 and 3.2 motivated most of the following results. Moreover, if P G (X) is a proper subgroup of G for all X ∈ L(G) − L(P (G)), then Proof. In order to prove the lower bound, it is enough to note from (2.1) that |P G (X)| once one uses the fact that P G (X) = G for every X ∈ L(P (G)). Now, since |G : P G (X)| = 1 for every X ∈ L(G) − L(P (G)), we get |G : P G (X)| ≥ p, that is, |P G (X)| ≤ |G| p . Therefore (5.1) is upper bounded by ≤ |L(P (G))||G| + |L(G)| − |L(P (G))| |G| p = (p − 1)|L(P (G))||G| p + |L(G)||G| p and the result follows.
Of course, D 8 satisfies the lower bound, but not the upper bound, of Theorem 5.1. Details can be deduced from the information of Example 3.2. This is to justify that Theorem 5.1 originates from evidences of computational nature. On the other hand, Example 3.2 shows also that Z(D 8 ) = P (D 8 ) ≃ C 2 . Then, when can we say that P (D 8 ) is noncyclic? The next two results concern this question.
Theorem 5.2. If P (G) is a nontrivial proper subgroup of a group G and pd(G) = 1 2 + |L(P (G))| 2 |L(G)| , then P (G) is noncyclic. Proof. By assumption we exclude the cases P (G) = G and P (G) = {1}, which are the extremal situations already known. Assume that P (G) is cyclic of prime order q ≥ p ≥ 2, where p is the smallest prime dividing |G|. We may apply the arguments of the proof of Theorem 4.3 and, noting that |L(P (G))| = 2, we find that |G| p |G| |L(G)| and then the inequality From this we derive the contradiction |L(G)| 2 ≤ 1, as at least {1} and G are contained in L(G). Therefore, P (G) cannot be cyclic of prime order and we may assume that P (G) is cyclic of order k ≥ 2. Now, we note that |L(P (G))| = |Div(k)|, where Div(k) is the set of all divisors of k. Here the argument we just used for q may still be applied. In fact we have and this would imply that |L(G)| ≤ |Div(k)| = |L(P (G))|, that is, L(G) ⊆ L(P (G)) and then L(G) = L(P (G)). This condition implies G = P (G), a contradiction.
The reader may note that Theorem 5.2 describes a very general situation, which cannot be reduced to those in Examples 3.1 and 3.2. In fact, looking at Example 3.1, P (D 6 ) = D 6 and so P (D 6 ) is not a proper subgroup of D 6 , and this means that one of the assumptions of Theorem 5. However, we may detect groups G with cyclic P (G). The following result shows this circumstance.

Computations for dihedral groups
We describe an instructive example, which correlates most of the notions which we have seen until now. Proposition 6.1. Let p be an odd prime. Then Proof. Noting that D 2p = C 2 ⋉ C p (see (4.1)) and that L(D 2p ) forms a diamond (as in Fig.III.3), we conclude that Z(D 2p ) is trivial and C D2p (C p ) = C p is the unique maximal normal subgroup of D 2p . Moreover D 2p is a P-group, because Q ∞ (D 2p ) = D 2p . Therefore a proper subgroup H of D 2p should be properly contained in P D2p (H) and necessarily P D2p (H) = D 2p . Thus, we find that pd(D 2p ) = 1.
From Proposition 6.1, pd(D 2p ) has a constant value for all odd primes, while sd(D 2p ) and d(D 2p ) are functions of p. This is an important difference of the permutability degree with respect to the subgroup commutativity degree and the commutativity degree. This reflects the fact that we are looking at permutizers in a group, and not at centralizers.

Some open questions
We end with a series of open questions. They arise naturally from the study of the present subject. The first is motivated by the families of dihedral groups, analyzed in Section 6 (and in other parts of the present paper). Most of these groups may be described in terms of product of groups. Open Question 7.1. Let G = N H be a product of a normal subgroup N by a subgroup H. What can be said about the permutability degree of G ?
It is well known that most of dihedral groups, generalized quaternion groups and semidihedral groups has this structure. They present further analogies in terms of central quotients: one, for instance, is that D 8 /Z(D 8 ) ≃ Q 8 and, roughly speaking, one can generalize this isomorphism to Q 2 4 = Q 16 , Q 2 5 = Q 32 and so on. The reader may refer to [15,16] for recent studies on these groups. Therefore: Open Question 7.2. What is the permutability degree of generalized quaternion groups and semidihedral groups ?
There is also another question, which is more general and may require some computational efforts. A classical result of Cayley allows us to embedd a group in a suitable symmetric group. The knowledge of symmetric groups plays in fact a fundamental role in several aspects of the theory of groups. Therefore: Open Question 7.3. What is the permutability degree of the symmetric group S n on n objects? Finally, the classification of (finite) simple groups may provide interesting aspects of study. For the (finite) simple (and almost simple) groups of sporadic type a lot is known about their subgroup lattices, see [7]. Therefore Open Question 7.4. What is the permutability degree of (finite) simple groups ?