The probability that a pair of group elements is autoconjugate

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1 Proc. Indan Acad. Sc. (Math. Sc.) Vol. 126, No. 1, February 2016, pp c Indan Academy of Scences The probablty that a par of group elements s autoconjugate MOHAMMAD REZA R MOGHADDAM 1,2,, ESMAT MOTAGHI 2 and MOHAMMAD AMIN ROSTAMYARI 1,3 1 Centre of Excellence n Analyss on Algebrac Structures, Ferdows Unversty of Mashhad and Department of Mathematcs, Khayyam Unversty, Mashhad, Iran 2 Internatonal Campus, Faculty of Mathematcal Scences, Ferdows Unversty of Mashhad, Mashhad, Iran 3 Quchan Unversty of Advanced Technology, Quchan, Iran * Correspondng author. E-mal: rezam@ferdows.um.ac.r; motagh.neda@yahoo.com; rostamyar@gmal.com MS receved 14 February 2014; revsed 11 July 2014 Abstract. Let g and h be arbtrary elements of a gven fnte group G. Theng and h are sad to be autoconjugate f there exsts some automorphsm α of G such that h = g α. In ths artcle, we construct some sharp bounds for the probablty that two random elements of G are autoconjugate, denoted by P a (G). ItsalsoshownthatP a (G) depends only on the autosoclnsm class of G. Keywords. Autoconjugate; autosoclnsm; autocommutator subgroup; autocentre. 1. Introducton 2010 Mathematcs Subject Classfcaton. Prmary: 20E45, 20B30; Secondary: 05A05, 05A16. Let G be a fnte group, then the autocommutator of the element g G and the automorphsm α n Aut(G) s defned to be [g,α]=g 1 g α = g 1 α(g). Usng ths defnton, the subgroup K(G) = [x,α]: x G, α Aut(G) s called the autocommutator subgroup of G. The concept of autocommutator subgroups has been already studed n [9, 16]. Also L(G) ={g G :[g,α]=1, α Aut(G)}, s called the autocentre of G. Clearly fα runs over the nner automorphsms of G, then K(G) and L(G) wll be the commutator subgroup G, and the centre Z(G) of G, respectvely. One notes that K(G) and L(G) are characterstc subgroups of G (see [8, 9] for more nformaton). 61

2 62 Mohammad Reza R Moghaddam et al. A group G acts on a non-empty set, f for every par (ω,g) G, the element ω g such that O 1.ω 1 G = ω; O 2.(ω g 1) g 2 = ω g 1g 2, for every g 1,g 2 G and all ω. Clearly ω G ={ω g g G} s the orbt of ω and G ω ={g G ω g = ω} s the stablzer of ω n G. It s easly seen that ω G =[G : G ω ] and = ω G G ω, for all ω n. Clearly Aut(G) acts on the group G and so the set of all elements, whch are autoconjugate to g s the autoconjugacy class of g and g Aut(G) = Aut(G) : C Aut(G) (g), n whch C Aut(G) (g) = Aut(G) g, whch s the stablzer of g n Aut(G). We defne P a (G) to be the probablty that two arbtrary elements g and h n G are autoconjugate. In fact P a (G) = {(g,h) G G : there exsts α Aut(G); h = gα } 2. Let g 1,g 2,...,g c be a complete set of representatves for the autoconjugacy classes of G. It s easy to see that P a (G) = 1 2 c =1 g Aut(G) 2. (1.1) In 1940, Hall [7] ntroduced the concept of soclnsm between two groups and t was extended to n-soclnsm, whch s an equvalent relaton among all groups and s weaker than somorphsm. In 1976, Leedham-Green and McKay [13] extended ths concept to sologsm wth respect to a gven varety of groups. There have been extensve studes n ths area of mathematcs (see [10, 11] for more nformaton). Now, we ntroduce the concept of autosoclnsm between two groups. DEFINITION 1.1 Let G and H be arbtrary groups and assume α : G/L(G) H/L(H), β : K(G) K(H) and γ : Aut(G) Aut(H) be homomorphsms such that the followng dagram s commutatve: G α γ Aut(G) L(G) H L(H) Aut(H) f 1 f 2 K(G) β K(H). (1.2) Then (α γ,β)s sad to be autohomoclnsm, so that G and H are called autohomoclnc. We also say that (α γ,β)s an autosoclnsm between G and H f α γ s surjectve and β and γ are njectve. In ths case, we say that G and H are autosoclnc and denoted by G a H (see [17] for more detal).

3 Probablty that a par of group elements s autoconjugate 63 Clearly, the above noton s a generalzed verson of homoclnsm, f one replaces Aut(X) by Inn(X), for any group X. In the present artcle, we construct some sharp upper and lower bounds forp a (G).We also show thatp a (G) s an nvarant of the autosoclnsm classes of G. Clearly, f the automorphsms are replaced by the nner automorphsms of G, then P a (G) = κ(g) s the probablty that two random elements g and h of G are conjugate and one obtans the results n [2] (see also [1, 3, 5, 6, 12, 15] for more nformaton on the subject). 2. Man results A useful general settng for ths probablty s gven by the relaton on the elements of a group G, defned by g h f h = g α,forsomeα Aut(G). Ths relaton naturally nduces an equvalence relaton on the autoconjugacy classes of G. In the followng theorem, we construct upper and lower bounds forp a (G). Theorem 2.1. Let G be a fnte non-trval group. Then 1 P a(g) K(G). Proof. By the defnton of autocommutator we may wrte g α = g[g,α], for all g G and α Aut(G) and hence g Aut(G) gk(g). Usng the defnton of P a (G), wemay wrte P a (G) = 1 2 g Aut(G) 1 2 gk(g). So t s clear thatp a (G) 1/ and the equalty holds exactly when G = Z 2. Thus, we obtan the followng: 1 P a(g) K(G). The followng result gves a conecton between the probablty of a group and ts factor group. Theorem 2.2. Let N be a non-trval characterstc subgroup of a fnte group G. Then P a (G) < P a (G/N). Proof. Let be the autoconjugacy relaton. It s clear that {(g,h) G G : g h} {(g,h) G G : gn hn}. The ncluson s strct, snce every automorphsm of G nduces an automorphsm of G/N, but the converse s not true, n general. On the other hand, one can easly check that {(g,h) G G : gn hn } = N 2 {(gn,hn) G/N G/N : gn hn }.

4 64 Mohammad Reza R Moghaddam et al. Thus we have {(g,h) G G : g h} 2 < {(gn,hn) N G N G : gn hn} N G, 2 and sop a (G) < P a (G/N). The next theorem wll be proved by the countng process. Theorem 2.3. For a fnte group G, fp a (G) < 7/4, then G s abelan. Proof. For a fnte non-abelan group G, f the non-autocentral classes of G and autoconjugacy classes of elements whch are not n autocentre of G have szes a 1,...,a c, then we have P a (G) = 1 2 c =1 g Aut(G) 2 = L(G) 2 + a a2 c 2. The sum a a2 c takes ts mnmum value, when a = 2 for all 1 c. Therefore a a2 c 22 ( L(G) )/2 and the equalty holds exactly when a = 2, for each. It follows that ( P a (G) 2 L(G) ). Snce G/Z(G) s non-cyclc, we have L(G) Z(G) /4. Hence P a (G) 7/4, and the equalty holds f and only f L(G) has ndex 4 n G and every non-autocentral conjugacy class has sze 2. The converse of the above theorem does not hold n general. Ths means that there are abelan groups for whch P a (G) 7/4, for example, P a (Z 8 ) = 11/32. Remark 2.4. Let c be the number of autoconjugacy classes of G. One can easly fnd a connecton between P a (G) and c, usng Cauchy Schwartz nequalty as follows: ( c ) 2 P a (G) g Aut(G) c = c 2 1 c, and the equalty holds exactly when g Aut(G) =1, for all g G and equvalently when 2. Moreover, f c s the number of autoconjugacy classes of G, one can mprove the upper and lower bounds stated n Theorem 2.1, as follows Theorem 2.5. Let G be a fnte group. Then P a (G) 4c 3 L(G) 2.

5 Probablty that a par of group elements s autoconjugate 65 Proof. Usng equaton (1.1), P a (G)= 1 c 2 g Aut(G) =1 g L(G) g Aut(G) 2 + It s clear that for every g G \ L(G) one has g Aut(G) 2 and hence P a (G) 1 4c 3 L(G) ( L(G) +4(c L(G) )) 2 2. g Aut(G) 2 g G\L(G) Clearly when every non-autocentral conjugacy class has sze 2, then the equalty holds. For example, one can easly calculate that P a (Z 8 ) = Now Theorem 2.1 mples 8 that 64 < On the other hand, the above theorem mples that 64 < Theorem 2.6. Let G be a fnte group. Then P a (G) 1 2 ( L(G) + Z(G)\L(G) Aut(G) Inn(G) + G\Z(G) Aut(G) 2. ). Proof. Usng defnton of P a (G), one has P a (G) = 1 2 g Aut(G) = g L(G) \Z(G) g Aut(G) + g Aut(G). g Z(G)\L(G) g Aut(G) Clearly, for every g Z(G) and φ x Inn(G), wehaveg φ x = g x = g. Thus Inn(G) C Aut(G) (g) and for all g Z(G) \ L(G), g Aut(G) = Aut(G) C Aut(G) (g) Aut(G) Inn(G). Also for every g G \ Z(G), one can easly check that C Aut(G) (g) > 2 and Therefore g Aut(G) = Aut(G) C Aut(G) (g) Aut(G). 2 P a (G) 1 ( 2 L(G) + Z(G)\L(G) Aut(G) ) Inn(G) + G\Z(G) Aut(G). 2 For example, P a (Z 5 ) = and one can easly see that Theorem 2.1 mples that 1, but usng Theorem 2.6, we obtan < <

6 66 Mohammad Reza R Moghaddam et al. In the proof of Theorem 2.3, we observed that ( P a (G) 2 L(G) ). (2.1) Usng the above nequalty, the followng example shows that the lower bound n Theorem 2.1 forp a (Z p n) s mproved. Example 2.7. (a) LetZ p n be the cyclc group of order p n, where p = 2 and n 1. Then φ : x x 2 s an automorphsm of Z p n. If there exsts <p n such that φ(x ) = x, then x 2 = x and so x = 1, whch contradcts the order of x. Thus L(Z p n) = 1 and hence by nequalty (2.1), P a (Z p n) 2pn 1 p 2n. (b) LetZ 2 n be the cyclc group of order 2 n,forn>1. Clearly, f r = 2t + 1 then the map φ : x x r s an automorphsm ofz 2 n. So r2 n 1 = (2t + 1)2 n 1 2 n 1 (mod 2 n ). Hence (x 2n 1 ) φ = x r2n 1 = x 2n 1. Now, f for each φ Aut(Z 2 n) and s N, (x s ) φ = x s, then x rs = x s and so x s(r 1) = 1, whch mples that 2 n s(r 1),.e. s = 2 n 1. Thus L(Z 2 n) ={e,x 2n 1 } and hence by nequalty (2.1), P a (Z 2 n) 2n 1 2 2n 1. Note that n the above examples, the equalty holds f and only f every autoconjugacy class has sze 2. For example, P a (Z 3 ) = 5/9. The followng lemma s helpful to calculate the probablty of autoconjugacy of elements n a drect product of groups, when the orders of drect factors are coprme. Lemma 2.8. Let H and K be two fnte groups such that ( H, K ) = 1. Then P a (H K) = P a (H) P a (K). Proof. By Theorem 2.1 of [16], we have Aut(H K) = Aut(H) Aut(K). Thus for every θ Aut(H K) there are α Aut(H) and β Aut(K) such that θ = (α,β). Accordng to the defnton of probablty, we have On the other hand, P a (H K) ={((h 1,k 1 ),(h 2,k 2 )) H K : θ Aut(H K); (h 2,k 2 ) = (h 1,k 1 ) θ, for all h 1,h 2 H and k 1,k 2 K}. (h 2,k 2 ) = (h 1,k 1 ) θ (h 2,k 2 ) = (h 1,k 1 ) (α,β) (h 2,k 2 ) = (h α 1,kβ 1 ) h 2 = h α 1,k 2 = k β 1. Thus (h 1,h 2 ) P a (H), (k 1,k 2 ) P a (K), and ths completes the proof.

7 Probablty that a par of group elements s autoconjugate 67 In the next result we show that for each autosoclnsm class A, there exsts a constant number k such that P a (G) = k/, for all groups G n A. For the specal case when takng the noton of soclnsm, see the man results n [4, 14]. Theorem 2.9. If G and H are autosoclnc fnte groups, then P a (G)=P a (H) H. Proof. By Defnton 1.1, and commutatvty of dagram (1.2), we have g Aut(G) = [g, Aut(G)] = f 1 (gl(g), Aut(G)), for all g G and f 1 s as n (1.3). Therefore P a (G)= 1 g Aut(G) = 1 [g, Aut(G)] = L(G) [g, Aut(G)] \L(G) = L(G) f 1 (g, Aut(G)). Smlarly, for the group H from the same dagram one gets P a (H) H = 1 h Aut(H) = L(H) H H h H As α and β are bjectons, we obtan f 1 (g, Aut(G)) = \L(G) = \L(G) h H\L(H) \L(G) h H\L(H) f 1 (g, Aut(G)) β f 2 (h, Aut(H)). On the other hand, G/L(G) and H/L(H) are somorphc and so P a (G)=P a (H) H, f 2 (h, Aut(H)). as requred. References [1] Alghamd A M and Russo F G, A generalzaton of the probablty that the commutator of two group elements s equal to a gven element, Bull. Iranan Math. Soc. 38 (2012) [2] Blackburn S R, Brtnell J R and Wldon M, The probablty that a par of elements of a fnte group are conjugate, J. London Math. Soc. 86(2) (2012) [3] Erfanan A, Lescot P and Rezae R, On the relatve commutatvty degree of a subgroup of a fnte group, Comm. Algebra 35 (2007) [4] Erfanan A, Rezae R and Russo F G, Relatve n-soclnsm classes and relatve nlpotency degree of fnte groups, Flomat 27 (2013)

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