Permutability Degrees of Finite Groups

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1 Filomat 30:8 (06), DOI 0.98/FIL60865O Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: htt:// Permutability Degrees of Finite Grous D. E. Otera a, F. G. Russo b a Institute of Mathematics and Informatics, Vilnius University, Vilnius, Lithuania b Deartment of Mathematics and Alied Mathematics, University of Cae Town, Cae Town, South Africa Abstract.Given a finite grou G, we introduce the ermutability degree of G, as d(g) P G (X), G where L(G) is the subgrou lattice of G and P G (X) the ermutizer of the subgrou X in G, that is, the subgrou generated by all cyclic subgrous of G that ermute with X L(G). The number d(g) allows us to find some structural restrictions on G. Successively, we investigate the relations between d(g), the robability of commuting subgrous sd(g) of G and the robability of commuting elements d(g) of G. Proving some inequalities between d(g), sd(g) and d(g), we correlate these notions.. Introduction All the grous of the resent aer are suosed to be finite. Given a grou G and its subgrou lattice L(G), the subgrou commutativity degree of G and the commutativity degree sd(g) {(H, K) L(G) L(G) HK KH} {(x, y) G G xy yx} d(g) G of G have been largely studied in the last years. Fundamental roerties and interesting generalizations of sd(g) can be found in [ 3,, 7 9], and for d(g) in [,, 6, 8 0, 4, 7 0, 4]. To study these notions, various ersectives have been considered in literature, because both measure theory and combinatorial techniques may be alied in order to get restrictions on the structure of a grou. The resent aer investigates a similar concet, the ermutability de ree of G d(g) P G (X) G 00 Mathematics Subject Classification. Primary 0P05; Secondary 06B3, 0D60. Keywords. Subgrou commutativity degree; ermutability degree; dihedral grous; robability of commuting airs. Received: May 04; Acceted: 03 May 05 Communicated by Francesco Belardo Dedication: To Eglė, Licia, Simona and Giovanni. addresses: daniele.otera@gmail.com (D. E. Otera), francescog.russo@yahoo.com (F. G. Russo)

2 D. E. Otera, F. G. Russo / Filomat 30:8 (06), and its connections with sd(g) and d(g). In the revious formula, the ermutizer P G (X) of a subgrou X of G is defined to be the subgrou generated by all cyclic subgrous of G that ermute with X, that is, P G (X) G X X. This means that X L(P G (X)) and X P G (X) if and only if X X for some G X. We concentrate on ermutizers because several classifications are available in literature on this toic. Recall that a grou G such that X P G (X) for every roer subgrou X of G is said to satisfy the ermutizer condition P, or briefly P rou. Therefore the ermutizer condition generalizes the well known normalizer condition (see [6]) and gives information on how the grou is near to be suersolvable. The study of ermutizers is not new and it is based on a series of fundamental contributions [3,, 3, 30] in the last 0 years. From [3, Corollary ], we know that for grous of odd order the ermutizer condition is equivalent of being suersolvable and actually, a comlete classification of P grous can be found in [3]. Now we may define the subgrou P(G) P G (H) H L(G) and correlate it with other subgrous of G. For instance, it is easy to check that the norm N(G) of G (see [6] for the roerties of N(G)) satisfies the following relation Z(G) C G (x) N(G) N G (H) P G (H) P(G). x G H L(G) H L(G) This relation emhasizes how P(G) is connected with other subgrous, 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 subgrou P(G) is also imortant because it allows us to maniulate the exression of d(g), for getting some analogies with sd(g) H L(G) C L(G) (H) and d(g) C G G (x), where P G (X) is the natural substitute of C L(G) (X) {Y L(G) YX XY} in [, 7] and of C G (x) {y G xy yx} in [, 8]. This maniulation of the exression of d(g) will allow us to detect whether P(G) is cyclic or not by looking only at the size of d(g). x G. Basic Proerties and Terminology Some of the following observations will be useful later on. Remark.. Since P G (X) is a subgrou of G, for all X L(G), and it always contains the trivial subgrou, then P G (X) G and also 0 < P G (X) G so that d(g) ]0, ]. Remark.. A grou G has d(g) if and only if the sum of all P G (X) for X L(G) is equal to G. By default, a quasihamiltonian grou G, that is, a grou in which every subgrou is ermutable, has d(g). A classification of quasihamiltonian grous can be found in [6, Theorems.4. and.4.6] and, roughly seaking, these grous are direct roducts of abelian grous by a coy of the quaternion grou of order 8. In articular, abelian grous have ermutability degree equal to. Another case in which the ermutability degree reaches is the following. Remark.3. A P grou G in which all roer subgrous are maximal has d(g). In such a case for all roer subgrous X of G one has X P G (X) G and so d(g). One might be temted to think that all P grous have ermutability degree equal to, but Examle 3. below shows this is false, and then the additional condition in which all roer subgrous are maximal cannot be omitted.

3 D. E. Otera, F. G. Russo / Filomat 30:8 (06), Now we rewrite the original exression of ermutability degree in the following more useful form. Since L(P(G)) is a sublattice of L(G), it turns out that d(g) P G G (X) + P G (X). (.) X L(P(G)) L(P(G)) Note also that a cyclic grou C (or better a quasihamiltonian grou Q) has sd(c) d(c) d(c) (or better sd(q) d(q) ). Therefore, relations between sd(g) and d(g) are meaningful when G is noncylic and nonquasihamiltonian (see Remark.). For the sake of comleteness, we recall some results of Beidleman and Heineken in [4]. The quasicenter Q(G) of G is the subgrou of G generated by all elements G such that K K, where K is an arbitrary subgrou of G. The subgrou Q(G) was introduced by Mukherjee and studied by several authors in the last years (see [4, 5, 5]), who investigated chains of quasicenters and relations with suersolvable grous. On the other hand, the hyerquasicenter of G, denoted by Q (G), is the largest term of the chain Q 0 (G) Q (G) Q(G)... Q i (G) Q i+ (G)... of normal subgrous of G, where, for any i 0, Q i+ (G)/Q i (G) Q(G/Q i (G)) and Q (G) i 0Q i (G). Recall that a normal subgrou N of G is said to be hyercyclically 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 hyercyclically embedded subgrou, which we denote Σ(G). More recisely, [4, Theorem ] shows that Σ(G) Q (G) is true for any grou G. Some interesting connections hold between P grous, P(G) and Q (G). For instance, [3, (3.),. 697] shows that a grou G is a P grou if and only if G/Σ(G) is a P grou. As a first consequence, a grou G is a P grou if and only if G/Q (G) is a P grou. As a second consequence, Z(G) Q(G) P(G) is true for any grou G. Furthermore, Q (G) P(G) if and only if P(G) Σ(G). 3. Examles Now we secify some of the revious notions for the symmetric grou S 3 on 3 objects. This will hel us to visualize analogies and differences between ermutability degrees, subgrou commutativity degrees and commutativity degrees. Examle 3.. The smallest nonabelian grou S 3 has L(S 3 ) {{}, S 3, A 3, H, K, L}, where A 3 (3) a, H () h, K (3) k, L (3) l. Noting that HK KH, HL LH, KL LK, one has P S3 ({}) P S3 (A 3 ) P S3 (S 3 ) S 3 ; P S3 (H) P S3 (K) P S3 (L) S 3 ; P(S 3 ) S 3. The fact that P S3 ({}) P S3 (A 3 ) P S3 (S 3 ) S 3 is clear, since we are dealing with ermutizers of normal subgrous. On the other hand, A 3 a P S3 (H) by definition. Now, H P S3 (H) is obvious. Then HA 3, which is a subgrou of S 3, should be contained in P S3 (H) and HA 3 H A 3 H A 3 6. This forces P S3 (H) to be equal to S 3. Similarly we get P S3 (K) P S3 (L) S 3. S 3 L A 3 H K Fig.III.. Hasse diagram of L(S3 ). {} It is interesting to note that an examle as easy as this has a lot of roerties in our ersective of study. The grou S 3 is suersovable by looking at the series {} 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 nilotent. Moreover,

4 D. E. Otera, F. G. Russo / Filomat 30:8 (06), S 3 has Z(S 3 ) {}, Σ(S 3 ) Q (S 3 ) Q (S 3 ) S 3, Q(S 3 ) A 3 and it is a P grou, since S 3 /Σ(S 3 ) {} is obviously a P grou. A direct calculation shows that 6 6 d(s 3 ) P S3 (X) P S3 (H) + P S3 (K) + P S3 (L) + P S3 (A 3 ) + P S3 (S 3 ) + P S3 ({}) 36, X L(S 3 ) then d(s 3 ) > sd(s 3 ) 5 6 > d(s 3), agreeing with the comutations in [7,.50] and [8, 9]. Another easy (but interesting) examle is the following. Examle 3.. The dihedral grou of order 8 is D 8 a, b a b 4, a ba b and has L(D 8 ) {{}, b, b, a, ba, b a, b 3 a, {, b, a, b a}, {, b, ba, b 3 a}, D 8 }. The normal subgrous are D 8, {}, B b, Z(D 8 ) b, M {, b, a, b a} and M {, b, ba, b 3 a}. Notice that H b a and K a are contained in M, while U ba and V b 3 a in M. D 8 M B M H K Z(D8 U ) V Moreover {} Fig.III.. Hasse diagram of L(D 8 ). 8 D 8 P D8 ({}) P D8 (D 8 ) P D8 ( b ) P D8 (Z(D 8 )) P D8 (M ) P D8 (M ), 4 M P D8 (H) P D8 (K), 4 M P D8 (V) P D8 (U) and Q(D 8 ) P(D 8 ) Z(D 8 ) < Σ(D 8 ) Q (D 8 ) D 8. In fact, D 8 Σ(D 8 ) is suersolvable and it is also a P grou, but nevertheless its ermutability degree is different from, because 8 0 d(d 8 ) P D8 (X) 6 D 8 + {, b, a, b a} + {, b, ba, b 3 a} 64. More recisely, X L(D 8 ) d(d 8 ) 5 8 < d(d 8) < sd(d 8) The value of d(d 8 ) can be found in [8] and that of sd(d 8 ) in [7]. This examle shows that there exist P grous with ermutability degree different from. Note that D 8 satisfies 8 D 8 < L(D 8 ) 0 but 8 {{}, D 8, b, {, b, ba, b 3 a}, {, b, a, b a}, b a, b, a } C L(D8 )( a ) Z L(D8 )( a ) {{}, b a, b, a } 4. Examles 3. and 3. illustrate a series of roblems for the comutation of the ermutability degree, arising from the nature of the subgrou lattice of the grous under consideration. We will come back to this oint later on. 4. General Proerties of the Permutability Degree We note that [8, Theorems.5, 3.3] shows that the commutativity degree is monotone. This is a well known roerty, which is due to the fact that we are dealing with a ositive monotone measure of robability. Similar situations can be found for sd(g) in [7, Proosition.4, Corollaries.5,.6,.7, Theorems 3.., 3..5] and in [9, 4, ]. For d(g) we have something similar.

5 Theorem 4.. Let H be a subgrou of a grou G. Then D. E. Otera, F. G. Russo / Filomat 30:8 (06), L(H) d(h) d(g). G : H Moreover, if P G (X) : P H (X) G : H for all X L(G), P(G) H and L(G) L(P(G)) L(P(G)), then d(g) L(H) d(h). In articular, L(H) L(H) d(h) d(g) G : H d(h). Proof. We start by roving the first inequality. Since P H (X) P G (X) for all X L(G), we have G d(g) P G (X) P H (X) X L(H) P H (X) d(h) H L(H) and the result follows. Now we rove the second inequality. Since by hyothesis, P G (X) : P H (X) G : H, we have P G (X) G : H P H (X) and so G d(g) P G (X) G : H P H (X) but, from (.), this means G : H X L(P(G)) P H (X) + L(P(G)) P H (X) and the inequality L(G) L(P(G)) L(P(G)) rovided by hyothesis, imlies G : H G : H X L(P(G)) X L(H) P H (X) + X L(P(G)) P H (X) G : H P H (X) G : H H L(H) d(h) from which we have d(g) L(H) d(h). X L(P(G)) P H (X) A classic slitting result for the roduct robability of two indeendent events is described by the following corollary. The roof may be generalized to finitely many factors, whose orders are airwise corime. Proosition 4.. Let G and H be two grous such that gcd( G, H ). Then d(g H) d(g) d(h). Proof. Given three grous A, B and C such that A B C, we know that N C (A B) N C (A) N C (B). This holds similarly for the ermutizers and it is easy to see that P C (A B) P C (A) P C (B). Now this fact and the assumtion gcd( G, H ) allow us to conclude that G H L(G H) X Y L(G H) P G H (X Y) G H L(H) P G (X) P H (Y). Y L(H)

6 D. E. Otera, F. G. Russo / Filomat 30:8 (06), The underlying roblem we deal with is the order of the subgrou lattices, which is hard to redict in general. If we concentrate on some grous arising from finite geometries, then the situation is more clear (dihedral grous, semidihedral grous and generalized quaternion grous were studied in [, 8,,, 7 9] from a similar ersective). Having in mind Examles 3. and 3., we observe from [6,. 6 9] and [, 7] that the dihedral grou D n x, y x y n, x yx y C C n x y (4.) of symmetries of a regular olygon with n edges has order n and slits in the semidirect roduct of a cyclic grou y C n of order n by a cyclic grou x C of order acting by inversion on C n. In articular, 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 [6] for this terminology) in between {} and D 6 and their number can be easily comuted. The following Fig.III.3 summarizes the information of Examle 3. in a more general situation. D C H H... H {} Fig.III.3. Hasse diagram of L(D ) with odd rime 3; H H... H C. From Figs.III., III. and III.3, it is clear that L(D ) has + roer subgrous, and this fact comes out from the following formula L(D n ) σ(n) + τ(n), (4.) where σ(n) and τ(n) are the sum and the number of all divisors of n (here n is arbitrary, not necessarily an odd rime), resectively. In articular, if n m is a ower of a rime (ossibly ) for some m 0, then the set of all divisors of m is Div( m ) {,,,..., m } so that σ( m ) m j0 j m+ and τ( m ) Div( m ) m +. (4.3) The reader has robably noted that we used the formula for the sum of a geometric series in the revious exression for σ( m ). Then we may conclude that L(D m) + m + m+ m + m+ +. (4.4) The next result shows an uer bound for d(g), when is of tye (4.4). Theorem 4.3. Let G be a noncyclic grou and the smallest rime divisor of G. If P(G) and m + m+ + for some m 0, then d(g) m+ + + (m 3) m m+ + (m + ) (m + ).

7 D. E. Otera, F. G. Russo / Filomat 30:8 (06), 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 assumtion. Without loss of generality we may assume P G (X) G for all X L(G). The minimality of imlies that P G (X) G for all X L(G). Of course, L(P(G)) and (.) becomes ( m m + m+ ) + G d(g) G + P G (X) L(P(G)) G + ( L(G) L(P(G)) ) G ( G G m m + m+ ) + G + ( ) G + G ( m m + m ) G ( m+ + ) + (m 3) m. This roves what we claimed. 5. Some Theorems of Structure The resent section is devoted to rove restrictions on P(G), arising from exact bounds for d(g), when G is an arbitrary grou. The evidences of Examles 3. and 3. motivated most of the following results. Theorem 5.. Let G be a grou (with d(g) ) and the smallest rime divisor of G. Then ( ) L(P(G)) + G G d(g). Moreover, if P G (X) is a roer subgrou of G for all X L(G) L(P(G)), then d(g) ( ) L(P(G)) +. Proof. In order to rove the lower bound, it is enough to note from (.) that G d(g) L(P(G)) G + P G (X) L(P(G)) G + ( L(P(G)) ), (5.) L(P(G)) where we have used in the last ste that P G (X). Then we continue ( G ) L(P(G)) + from which we get d(g) ( G ) L(P(G)) + G ( G ) L(P(G)) G + G. Now we rove the uer bound. Formula (.) becomes again (5.) G d(g) L(P(G)) G + P G (X) L(P(G)) once one uses the fact that P G (X) G for every X L(P(G)). Now, since G : P G (X) for every X L(G) L(P(G)), we get G : P G (X), that is, P G (X) G. Therefore (5.) is uer bounded by L(P(G)) G + ( L(P(G)) ) G and the result follows. ( ) L(P(G)) G + G

8 D. E. Otera, F. G. Russo / Filomat 30:8 (06), Of course, D 8 satisfies the lower bound, but not the uer bound, of Theorem 5.. Details can be deduced from the information of Examle 3.. This is to justify that Theorem 5. originates from evidences of comutational nature. On the other hand, Examle 3. shows also that Z(D 8 ) P(D 8 ) C. Then, when can we say that P(D 8 ) is noncyclic? The next two results concern this question. Theorem 5.. If P(G) is a nontrivial roer subgrou of a grou G and d(g) + L(P(G)), then P(G) is noncyclic. Proof. By assumtion we exclude the cases P(G) G and P(G) {}, which are the extremal situations already known. Assume that P(G) is cyclic of rime order q, where is the smallest rime dividing G. We may aly the arguments of the roof of Theorem 4.3 and, noting that L(P(G)), we find that d(g) + L(P(G)) + + G G ( ) ( + ) G + G G G and then the inequality which means + L(G) + ( ) ( ) +. From this we derive the contradiction, as at least {} and G are contained in L(G). Therefore, P(G) cannot be cyclic of rime order and we may assume that P(G) is cyclic of order k. 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 alied. In fact we have then which means + Div(k) + Div(k) Div(k) G G ( Div(k) ) + G G + Div(k) ( Div(k) + Div(k) ) G G L(G) + Div(k) ( ) ( ) Div(k) ( ) Div(k) + and this would imly that Div(k) L(P(G)), that is, L(G) L(P(G)) and then L(G) L(P(G)). This condition imlies G P(G), a contradiction. The reader may note that Theorem 5. describes a very general situation, which cannot be reduced to those in Examles 3. and 3.. In fact, looking at Examle 3., P(D 6 ) D 6 and so P(D 6 ) is not a roer subgrou of D 6, and this means that one of the assumtions of Theorem 5. is not satisfied. On the other hand, P(D 8 ) Z(D 8 ) is a nontrivial roer subgrou of D 8 (see Examle 3.), but 4 5 d(d 8) , and in fact P(D 8 ) is cyclic. Once again, Theorem 5. can not be alied. These two examles show that we cannot strengthen further Theorem 5.. However, we may detect grous G with cyclic P(G). The following result shows this circumstance. Theorem 5.3. Let P(G) be a nontrivial roer subgrou of a grou G with d(g) 4 5 divisor of G. If 4 G 5 5 G 5, then P(G) is cyclic of rime order. and be the smallest rime

9 D. E. Otera, F. G. Russo / Filomat 30:8 (06), Proof. From the lower bound of Theorem 5., we have d(g) 4 ( 5 ) L(P(G)) + 4 G G 5 4 G 5 5 G G G L(P(G)) G G G 4 G 5 5 G 5 L(P(G)). L(P(G)) We conclude that L(P(G)), hence L(P(G)). This forces P(G) to be cyclic of rime order. 6. Comutations for Dihedral Grous We describe an instructive examle, which correlates most of the notions which we have seen until now. Proosition 6.. Let be an odd rime. Then d(d ) > sd(d ) > d(d ). Proof. Noting that D C C (see (4.)) and that L(D ) forms a diamond (as in Fig.III.3), we conclude that Z(D ) is trivial and C D (C ) C is the unique maximal normal subgrou of D. Moreover D is a P grou, because Q (D ) D. Therefore a roer subgrou H of D should be roerly contained in P D (H) and necessarily P D (H) D. Thus, we find that d(d ). On the other hand, we may secialize the formula given in [7, Theorem 3..], where sd(d ) τ() + τ()σ() + () (τ() + σ()), () is the arithmetic function in [7, Eq. 0,.54]. From (4.3) we deduce τ() and σ() + so that sd(d ) 4 + ( + ) Now [0, Remark 4.] shows that d(d ) > ( )( + ) and for all odd rimes we have > > ( ) > ( + 3)( ) > The result follows. From Proosition 6., d(d ) has a constant value for all odd rimes, while sd(d ) and d(d ) are functions of. This is an imortant difference of the ermutability degree with resect to the subgrou commutativity degree and the commutativity degree. This reflects the fact that we are looking at ermutizers in a grou, and not at centralizers.

10 D. E. Otera, F. G. Russo / Filomat 30:8 (06), Some Oen Questions We end with a series of oen questions. They arise naturally from the study of the resent subject. The first is motivated by the families of dihedral grous, analyzed in Section 6 (and in other arts of the resent aer). Most of these grous may be described in terms of roduct of grous. Oen Question 7.. Let G NH be a roduct of a normal subgrou N by a subgrou H. What can be said about the ermutability degree of G? It is well known that most of dihedral grous, generalized quaternion grous and semidihedral grous has this structure. They resent further analogies in terms of central quotients: one, for instance, is that D 8 /Z(D 8 ) Q 8 and, roughly seaking, one can generalize this isomorhism to Q 4 Q 6, Q 5 Q 3 and so on. The reader may refer to [5, 6] for recent studies on these grous. Therefore: Oen Question 7.. What is the ermutability degree of generalized quaternion grous and semidihedral grous? There is also another question, which is more general and may require some comutational efforts. A classical result of Cayley allows us to embed a grou in a suitable symmetric grou. The knowledge of symmetric grous lays in fact a fundamental role in several asects of the theory of grous. Therefore: Oen Question 7.3. What is the ermutability degree of the symmetric grou S n on n objects? Finally, the classification of (finite) simle grous may rovide interesting asects of study. For the (finite) simle (and almost simle) grous of soradic tye a lot is known about their subgrou lattices, see [7]. Therefore Oen Question 7.4. What is the ermutability degree of (finite) simle grous? Acknowledgements The first author was artially suorted by GNSAGA of INdAM (Italy). The research of the second author is suorted in art by NRF (South Africa) for the Grant No and in art from the Launching Grant No of the University of Cae Town (South Africa). He also thanks Universidade Federal do Rio de Janeiro (Rio de Janeiro, Brazil), where the resent subject was originally investigated via the roject CAPES No. 06/03. Finally, we thank the editor for his valuable work and the referee for the questions in Section 7. References [] A. M. Alghamdi, D. E. Otera and F. G. Russo, On some recent investigations of robability in grou theory, Boll. Mat. Pura Al. 3 (00), [] A. M. Alghamdi and F. G. Russo, A generalization of the robability that the commutator of two grou elements is equal to a given element, Bull. Iran. Math. Soc. 38 (0), [3] J. C. Beidleman and D. J. Robinson, On finite grous satisfying the ermutizer condition, J. Algebra 9 (997), [4] J. C. Beidleman and H. Heineken, On the hyerquasicenter of a grou, J. Grou Theory 4 (00), [5] J. C. Beidleman, H. Heineken and F. G. Russo, Generalized hyercenters in infinite grous, Asian-Eur. J. Math. 4 (0), 30. [6] S. Blackburn, J. R. Britnell and M. Wildon, The robability that a air of elements of a finite grou are conjugate, J. London Math. Soc. 86 (0), [7] T. Connor and D. Leemans, An atlas of subgrou lattices of finite almost simle grous, Ars Math. Contem. 8 (05), [8] A. Erfanian, P. Lescot and R. Rezaei, On the relative commutativity degree of a subgrou of a finite grou, Comm. Algebra 35 (007), [9] A. Erfanian, R. Rezaei and F. G. Russo, Relative n-isoclinism classes and relative nilotency degree of finite grous, Filomat 7 (03), [0] A. Erfanian and F. G. Russo, Probability of the mutually commuting n-tules in some classes of comact grous, Bull. Iranian Math. Soc. 34 (008), [] M. Farrokhi and F. Saeedi, Factorization numbers of some finite grous, Glasg. Math. J. 54 (0), [] M. Farrokhi, Factorization numbers of finite abelian grous, Int. J. Grou Theory (03), 8.

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