Probability that an Autocommutator Element of a Finite Group Equals to a Fixed Element

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1 Filomat 3:20 (207), Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: Probability that an Autocommutator Element of a Finite Group Equals to a Fixed Element Zohreh Sepehrizadeh a, Mohammad Reza Rismanchian a a Department of Pure Mathematics, Shahrekord University, Shahrekord, Iran Abstract. In this article we introduce a formula for the probability which an autocommutator element of a finite group G, equals to a fixed element of G and derive some properties of this formula. Moreover, we obtain a lower bound and an upper bound for this probability in the special cases. This generalizes some results of Das et al. in 200 and Moghaddam et al. in 20.. Introduction The concept of commutativity degree started by Gustafson in 975, which is probability that two elements of a finite group G commute. In this article G denotes a finite group, H a subgroup of G, and an element of G. In [6], Moghaddam et al. have considered the probability Paut(H, G) for an element randomly chosen of H, which is fixed by an automorphism randomly chosen of Aut(G). On the other hand, in [2], Das and Nath have studied the probability Pr (H, G) = where [x, y] = x y xy. We will study the ratio P aut(g) = {(x, y) H G : [x, y] = }, G {(x, α) G Aut(G) : [x, α] = }, G where [x, α] = x x α, is called autocommutator element of G (see also [7]). Moreover, we extend some of the results obtained in [2]. We also develop and characterize a formula for probability of the pair (H, G), which generalizes the formula for Pr (H, G) given in [2]. Note that if H = G then Paut(H, G) = P aut(g), which coincides with autocommutativty degree Paut(G) of G, if we take =, the identity element of G. We note that this case is treated in [7]. It may be recalled that Paut(G) = k where k denotes the number of disjoint orbits of G under the group automorphism Aut(G) G 200 Mathematics Subject Classification. 20B40, 20D45, 20P05. Keywords. Autocommutativity degree, Relative autocommutativity degree, Autocommutators. Received: 0 August 204; Revised: 2 April 207; Accepted: December 207 Communicated by Dragan S. Djordjević addresses: zohreh.sepehri@gmail.com (Zohreh Sepehrizadeh), rismanchian33@gmail.com, rismanchian@sci.sku.ac.ir (Mohammad Reza Rismanchian)

2 Z. Sepehrizadeh, M.R. Rismanchian / Filomat 3:20 (207), (see also [3 5] for commutativity degree and generalization). Among the other results, we particularly determine an upper bound for relative autocommutativity degree of a finite group. Let G be a group and Aut(G) be the full automorphisms group of G, for all α Aut(G) and G, we have the following map Aut(G) G G (α, ) α. For an element G, the set of automorphisms {α Aut(G) α = } denoted by C Aut(G) ( ) is a subgroup of Aut(G) and the equivalence classes { α α Aut(G) } denoted by Orb Aut(G) ( ). Let H be a subgroup of G, we introduce two subgroups of Aut(G) and H as follows, respectively, C Aut(G) (H) = {α Aut(G) h α = h, h H }, C H (Aut(G)) = {h H h α = h, α Aut(G) }. In this notation we introduce a subgroup of G as follows, L(G) = { G [, α] =, α Aut(G) }, which is called absolute centre of G, see also [6]. 2. Main Results In this section we first give the following definition which generalizes definitions of Das et al. and Moghaddam et al., see [2, 6]. Definition 2.. Let G be a finite group, H a subgroup of G and an element of G then the probability P aut(h, G) is defined as follows: {(x, α) H Aut(G) : [x, α] = } P aut(h, G) =. where [x, α] = x x α. Let [H, Aut(G)] be the subgroup of G generated by autocommutators [x, α] = x x α with x H and α Aut(G). Also, for the sake of simplicity let us write P aut(h, G) = Paut(H, G). Clearly, Paut(H, G) = [H, Aut(G)] = {}, and P aut(h, G) = 0 { [x, α] : x H, α Aut(G) }. It is also easy to see that if C Aut(G) (x) = {} for all x H {} then Paut(H, G) = +. We now derive a computing formula, as generalization of Theorem 2.3 of [2], which plays a key role in the study of P aut(h, G). Theorem 2.2. The following statement holds P aut(h, G) = where Orb AG (x) = { x α : α Aut(G) }. C Aut(G) (x) = Orb AG (x),

3 Z. Sepehrizadeh, M.R. Rismanchian / Filomat 3:20 (207), Proof. We have {(x, α) H Aut(G) : x x α = } = {x} T x, where T x = {α Aut(G) : x x α = }. Note that, for any x H, we have T x x Orb AG (x). Let T x for some x H. Fix an element α 0 T x, then α α α defines an one to one correspondence 0 between the set T x and C Aut(G) (x). This means that T x = C Aut(G) (x). Thus, we have {(x, α) H Aut(G) : x x α = } = T x = C Aut(G) (x). The first equality in the theorem follows from Definition 2.. For the second equality, consider the action of Aut(G) on G by the above discussion. Then, for all x G, we have Orb AG (x) = Aut(G) : Stab(x) = C Aut(G) (x). This completes the proof. As an immediate consequence, we have the following generalization of the well-known formula Paut(G) = k, where k is the number of the disjoint orbits of G under the group automorphism Aut(G) (see also [6, 8]). G Corollary 2.3. If H is a characteristic subgroup of G, then Paut(H, G) = k(h), where k(h) is the number of the disjoint orbits of H under the group automorphism Aut(G). Proof. Note that Aut(G) acts on H, for all α Aut(G) and x H, we have the following map Aut(G) H H (α, x) x α. The orbit of any element x H under this action is given by Orb AG (x), hence H is the disjoint union of these classes. Therfore, we have Paut(H, G) = Orb AG (x) = k(h). Note that for =, the condition x Orb AG (x) is superfluous. Lemma 2.4. (J. H Christopher and L. R. Darren []) Let G and G 2 be two groups such that cd( G, G 2 ) =, then Aut(G G 2 ) Aut(G ) Aut(G 2 ). Remark 2.5. Let G = G G 2, H = H H 2, and for all h i H i, i =, 2. One can easily see that any automorphism α of Aut(G i ), (i =, 2) may be extended to an automorphism α e of Aut(G), in such a way that ( 2 ) αe = α 2, for all, 2 G. We denote all such extended automorphisms in Aut(G) by Aut(G e ), which are one-to-one correspondence i with the ones in Aut(G i ). So it is clear that Aut(G e i ) = Aut(G i), for i =, 2 and Aut(G e ) Aut(Ge 2 ) =< id G >, Aut(G e )Aut(Ge 2 )C Aut(G)(h h 2 ) Aut(G). Hence Aut(G e )Aut(Ge 2 ) C Aut(G)(h h 2 ) Aut(G e )Aut(Ge 2 ) C Aut(G)(h h 2 ) = Aut(Ge ) Aut(Ge 2 ) C Aut(G)(h h 2 ) Aut(G e )Aut(Ge 2 ) C Aut(G)(h h 2 ) = Aut(G e ) Aut(Ge 2 ) C Aut(G)(h h 2 ) Aut(G e ) C Aut(G)(h h 2 ) Aut(G e 2 ) C Aut(G)(h h 2 ) = Aut(Ge ) Aut(Ge 2 ) C Aut(G)(h h 2 ) C Aut(G )(h ) C Aut(G2 )(h 2 ),

4 which implies that C Aut(G)(h h 2 ) Z. Sepehrizadeh, M.R. Rismanchian / Filomat 3:20 (207), C Aut(G )(h ) C Aut(G2 )(h 2 ) Aut(G e ) Aut(Ge 2 ), see also [6]. The following theorem is a generalization of the work of Moghaddam et al. which in turn is similar to Theorem 2. of [6]. Theorem 2.6. Let H and H 2 be subgroups of the finite groups G and G 2, respectively. Let G and 2 G 2. Then P (, 2 )aut(h H 2, G G 2 ) P aut(h G )P 2 aut(h 2 G 2 ). In particular, the equality holds when cd( G, G 2 ) =. Proof. Put G = G G 2, H = H H 2 and = (, 2 ), for all i G i and h i H i, i =, 2. Then from Theorem 2.2 and the above remark, it follows that P aut(h, G) = = = h H h Orb AG (x) H H 2 H H 2 H C Aut(G) (h) h h 2 H h i i Orb AGi (h i ) C Aut(G) (h h 2 ) h H h 2 H 2 h Orb AG (h ) h 2 2 Orb AG2 (h 2 ) h H h Orb AG (h ) C Aut(G )(h ) Aut(G ) = P aut(h, G )P 2 aut(h 2, G 2 ). H 2 where i =, 2. C Aut(G )(h ) C Aut(G2 )(h 2 ) Aut(G ) h 2 H 2 h 2 2 Orb AG2 (h 2 ) Aut(G 2 ) C Aut(G2 )(h 2 ) Aut(G 2 ) Now, using the assumption cd( G, G 2 ) = and Lemma 2.4 the equality yeilds. Next, we present the following proposition which is similar to Proposition 3. in [2]. Proposition 2.7. If then (i) P aut(h, G) 0 = P aut(h, G) C H(Aut(G)) C Aut(G) (H), L(G) (ii) P aut(g) 0 = P aut(g) G. Proof. Let = [x, α] for some (x, α) H Aut(G). Since, we have x C H (Aut(G)) and α C Aut(G) (H). Consider the left coset T (x,α) = (x, α)c H (Aut(G)) C Aut(G) (H) of C H (Aut(G)) C Aut(G) (H) in H Aut(G). Clearly, T (x,α) = C H (Aut(G)) C Aut(G) (H), and [a, β] = for all (a, β) T (x,α). This proves part (i). Similarly, part (ii) follows with H = G. Now by using Theorem 2.2, we are able to generalize Proposition 3.2 of Das et al. as follows: Proposition 2.8. Let H be a subgroup of group G and G then with equality if and only if =. P aut(h, G) Paut(H, G)

5 Proof. By Theorem 2.2, we have P aut(h, G) = Z. Sepehrizadeh, M.R. Rismanchian / Filomat 3:20 (207), C Aut(G) (x) C Aut(G) (x) = Paut(H, G). Clearly, the equality holds if and only if x Orb AG (x) for all x H, that is the equality holds if and only if =. As a generalization of Proposition 3.3 of [2], we establish the following theorem. Theorem 2.9. Let p be the smallest prime dividing, and. Then, P aut(h, G) C H(Aut(G)) p Proof. Without loss of generality, we may assume that C H (Aut(G)) H. Let x H be such that x Orb AG (x). Then, Since, we have x C H (Aut(G)) and Orb AG (x) >. Since Orb AG (x) = Aut(G) : Stab(x) = C Aut(G) (x), so Orb AG (x) is a divisor of. Therefore, Orb AG (x) p. Hence, by Theorem 2.2, we have P aut(h, G) p C H(Aut(G)) < p p, which completes the proof. < p. Proposition 2.0. Let H and H 2 be subgroups of a group G and H is contained in H 2, then P aut(h, G) H 2 : H P aut(h 2, G). The equality holds if and only if x Orb AG (x) for all x H 2. Proof. By Theorem 2.2, we have H P aut(h, G) = C Aut(G) (x) C Aut(G) (x) 2 = H 2 P aut(h 2, G). The condition for equality follows immediately. Immediate consequence of the above proposition is as follows: Corollary 2.. with equality if and only if = and H = G. P aut(h, G) G : H Paut(G),

6 Z. Sepehrizadeh, M.R. Rismanchian / Filomat 3:20 (207), Acknowledgment The authors would like to thank the referees for their useful comments that helped to improve the presentation of the article. References [] J. H. Christopher, L. R. Darren, Automorphism of finite abelian groups, American Mathematical Monthly 4 (2007) [2] A. K. Das, R. K. Nath, On generalized relative commutativity degree of a finite group, International Electronic Journal of Algebra 7 (200) [3] A. K. Das, R. K. Nath, On generalized commutativity degree of a finite group, Rocky Mountain Journal of Mathematics 4 (6) (20) [4] W. H. Gustafson, What is the probability that two group elements commute?, American Mathematical Monthly 80 (973) [5] P. Lescot, Isoclinism classes and commutativity degrees of finite groups, Journal of Algebra 77 (995) [6] M. R. R. Moghaddam, F. Saeedi, E. Khamseh, The probability of an automorphism fixing a subgroup element of a finite group, Asian-European Journal of Mathematics 4 (2) (20) [7] M. R. Rismanchian, Z. Sepehrizadeh, Autoisoclinism classes and autocommutativity degrees of finite groups, Hacettepe Journal of Mathematics and Statistics 44 (4) (205) [8] G. J. Sherman, What is the probability an automorphism fixes a group element?, American Mathematical Monthly 82 (975)