Bulleti of the Iraia Mathematical Society Vol. 39 No. 2 203), pp 27-280. FINITE GROUPS WITH THREE RELATIVE COMMUTATIVITY DEGREES R. BARZGAR, A. ERFANIAN AND M. FARROKHI D. G. Commuicated by Ali Reza Ashrafi Abstract. For a fiite group G ad a subgroup H of G, the relative commutativity degree of H i G, deoted by dh, G), is the probability that a elemet of H commutes with a elemet of G. Let DG) = {dh, G) : H G} be the set of all relative commutativity degrees of subgroups of G. It is show that a fiite group G admits three relative commutativity degrees if ad oly if G/ZG) is a o-cyclic group of order pq, where p ad q are primes. Moreover, we determie all the relative commutativity degrees of some kow groups.. Itroductio If G is a fiite group, the the commutativity degree of G, deoted by dg), is the probability that two radomly chose elemets of G commute. The commutativity degree first studied by Gustafso [4] ad it was show that dg) 5 8 for every o-abelia fiite group G ad equality holds precisely whe G/ZG) = Z 2 Z 2. Let H be a subgroup of G. Erfaia et al. i [] geeralized the commutativity degree of G by cosiderig the relative commutativity MSC200): Primary: 20P05; Secodary: 20E45. Keywords: Commutativity degree, relative commutativity degree, isocliism, relative isocliism. Received: 8 July 20, Accepted: 2 April 202. Correspodig author c 203 Iraia Mathematical Society. 27
272 Barzegar, Erfaia ad Farrokhi degree of H i G, deoted by dh, G), as the probability that a elemet of H commutes with a elemet of G that is {h, g) H G : [h, g] = } dh, G) =. G Also they have give several lower ad upper bouds for the relative commutativity degree dh, G). We refer the reader to [, 4, 5] for more details. Now let DG) = {dh, G) : H G}. It is obvious that DG) = if ad oly if G is a abelia group. Also, it is easy to see that there is o fiite group G with DG) = 2 see Lemma 2.4). We ited to study the set DG) ad classify all fiite groups G with three commutativity degrees. We will show that if G is a fiite group, the DG) = 3 if ad oly if G/ZG) is a group of order pq for some primes p ad q. Moreover, the umber of relative commutativity degrees will be computed for some classes of fiite groups icludig dihedral groups, geeralized quaterio groups ad quasi-dihedral groups. The motivatio to the research is the Farrokhi s classificatio of fiite groups with two subgroup ormality degrees i [3]. For further details o this topic, we refer the iterested reader to [6]. 2. Prelimiary results We begi with some basic lemmas. Lemma 2.. Let G be a fiite group ad H K G. The dk, G) dh, G) ad the equality holds if ad oly if K = HC K g) for all g G. Proof. Sice g H g K, we have C Kg) K dk, G) = C K g) G K G g G C Hg) g G for each g G. Hece C H g) Also dk, G) = dh, G) if ad oly if C Kg) K which is equivalet to K = HC K g) for all g G. Note that for ay subgroup H G, we have dh, G) = C H g). G g G = C Hg) = dh, G). for all g G, Lemma 2.2. Let G be a o-abelia fiite group ad x G \ ZG). The d x, G), dg).
Relative commutativity degrees 273 Proof. Clearly d x, G) because x is ot cetral. Suppose that d x, G) = dg), the by Lemma 2., G = x C G y) for all y G. I particular G = x C G x) = C G x), which implies that x ZG), a cotradictio. From the above lemma the followig result ca be obtaied immediately. Corollary 2.3. If G is a o-abelia fiite group, the DG) 2. I the sequel we shall discuss fiite groups with three relative commutativity degrees. Lemma 2.4. Let G be a o-abelia fiite group ad suppose that DG) = {, d, dg)}. If H is a subgroup of G such that dh, G) = d, the H is abelia. Proof. Let h H \ ZH), the by Lemma 2.2, d h, G) dg). Thus d h, G) = dh, G) ad by Lemma 2., H = h C H g) for all g G. Now by replacig g by h, we coclude that h ZH), a cotradictio. Lemma 2.5. Let G be a fiite group with DG) = 3. The C G x) is a abelia maximal subgroup of G, for all x G \ ZG). Proof. Let x G \ ZG). If C G x) is a o-abelia group, the by Lemmas 2.4 ad 2., we have G = C G x)c G g) for all g G. I particular, G = C G x)c G x) = C G x) ad hece x ZG), which is a cotradictio. Now let M be a maximal subgroup of G cotaiig C G x). If M is o-abelia, the by Lemmas 2.4 ad 2., G = MC G x) so G = M, a cotradictio. Thus M is abelia ad cosequetly M = C G x), as required. Lemma 2.6. If H is a subgroup of G, the dhzg), G) = dh, G). Proof. The proof is straightforward. Lemma 2.7. Let G be a fiite group with DG) = 3. The the followig assertios are true: i) if x is a p-elemet, the x p ZG); ii) if x, y G\ZG) are p-elemet ad q-elemet, respectively, the p q) G = qp ) C G x) pq ) C G y). I particular, C G x) = C G y) if ad oly if p = q.
274 Barzegar, Erfaia ad Farrokhi Proof. i) Let x G \ ZG) be a p-elemet of order p. We proceed by iductio o. Cearly the result holds for = 0,. Suppose that the result holds for. The x p = p ad by hypothesis x p2 ZG). If x p ZG), the by Lemma 2.2, d x, G) = d x p, G). O the other had, d x, G) = p G p 2 G +p p 2 ) C G x p ) +p p ) C G x) ) ad d x p, G) = from which we obtai p G p 2 G + p p 2 ) C G x p ) ), p = [G : C G x)] + p )[C G x p ) : C G x)]. Hece [G : C G x)] = [C G x p ) : C G x)] = that is x ZG), cotradictig the hypothesis. ii) By Lemma 2.2, d x, G) = d y, G). By part i) we have ad d x, G) = p m G pm G + p m p m ) C G x) ) d y, G) = q G q G + q q ) C G y) ), from which the result follows. The rest of the proof is a direct cosequece of the last two equatios. 3. Mai results We are ow i a positio to give the mai theorems. The ilpotet ad o-ilpotet cases are discussed separately. Theorem 3.. Let G be a fiite ilpotet group. The DG) = 3 if ad oly if G/ZG) = Z p Z p. I particular, DG) = {, 2p, p2 +p }. p 2 p 3 Proof. Sice G is ilpotet, so G = P P 2 P, where P i is the Sylow p i -subgroup of G. Clearly DG) = DP )DP 2 ) DP ) DP ) DP 2 ) DP ). Sice DG) = 3, there exists a Sylow p i -subgroup P = P i such that DP ) = DG) ad P j is abelia for each j i. Let x, y P such that xy yx, M = C P x) ad N = C P y). The by Lemma 2.5i),
Relative commutativity degrees 275 M ad N are abelia maximal subgroups of P. Clearly P = MN ad M N = ZP ). Hece P ZP ) = MN M N = M M N N M N = p2. Coversely, let H G be a proper o-cetral subgroup of G such that dh, G) dg). By Lemma 2.6, dh, G) = dhzg), G) ad we may assume that ZG) H. The H/ZG) = Z p ad we have dh, G) = G = p ZG) = 2p p 2. h H C G h) = h H h G ) ZG) + p ) ZG) p Similarly, it ca be easily show that dg) = p2 +p p 3 ad cosequetly DG) = {, 2p, p2 +p }. p 2 p 3 Theorem 3.2. Let G be a fiite o-ilpotet group. The DG) = 3 if ad oly if G/ZG) is a o-cyclic group of order pq, where p ad q are distict primes. I particular, DG) = {, p + q pq, p + }, q 2 pq 2 wheever p > q. Proof. Sice G is ot ilpotet, there exist a p-elemet x ad a q-elemet y p q) such that xy yx. Clearly we may assume that q is the smallest prime dividig G/ZG). By Lemma 2.5, M = C G x) ad N = C G y) are differet abelia maximal subgroups of G. Moreover, M N = ZG) ad by Lemma 2.7 3.) p q) G = qp ) M pq ) N. Note that by Lemma 2.7ii), M N so that M ad N are ot cojugate. Let : G G/ZG) be the atural homomorphism. If M ad N are o-ormal subgroups of G, the N G M) = M, N G N) = N ad the
276 Barzegar, Erfaia ad Farrokhi cojugates of M ad N all have trivial itersectio. Thus G M g N g g G g G = + [G : M] M ) + [G : N] N ) + G 2 + G 2 > G, which is a cotradictio. Thus M or N is a ormal subgroup of G, which implies that G = MN. If M, N G, the G = M N is abelia ad hece G is ilpotet, a cotradictio. Therefore G is a Frobeius group with M ad N as its Frobeius kerel ad complemet i some order. Moreover, gcd M, N ) =. From the equatio 3.) it follows that M divides pq ) N, which implies that M divides p. Hece M = p ad, usig the equatio 3.) oce more, we obtai N = q. Therefore G = pq, as required. Coversely, suppose that G is a fiite o-ilpotet group such that G/ZG) = pq for some primes. Clearly G/ZG) is o-abelia, p q ad we may assume that p > q. Let H be a proper o-cetral subgroup of G. The by Lemma 2.6, we may assume that ZG) H. Thus H/ZG) = Z p or Z q. I the first case dh, G) = ad i the secod case dh, G) = ZG) + p ) ZG) q ZG) + q ) ZG) p ) = p + q pq ) = p + q pq. O the other had G/ZG) is a Frobeius group with Frobeius kerel ad complemet isomorphic to cyclic groups of orders p ad q, respectively. Thus dg) = G ZG) + p ) ZG) ) q + pq p) ZG) p = p + q 2 pq 2. Therefore DG) = {, p + q pq, p + q 2 pq 2 } ad the proof is complete.
Relative commutativity degrees 277 4. Examples This subsectio is devoted to the determiatio of the set of all relative commutativity degrees of some classes of isocliic fiite groups. First, we compute the set of all relative commutativity degrees of dihedral groups, the we apply isocliism betwee groups to obtai the relative commutativity degrees for some other classes of groups. Theorem 4.. Let G = D 2 = a, b : a = b 2 =, a b = a be the dihedral group of order 2. The { 2τ), odd, DG) = 2kτm), = 2 k m, k ad m odd. where τm) is the umber of divisors of m. Proof. It is kow that a arbitrary subgroup of G = D 2 has the form a k, a t b or a k, a l b, where t = 0,,, k ad l = 0,, 2,..., k. We proceed i two steps. First suppose that is odd. The d a k, G) = a k h G = k k + a ki ) G h a k i= = k + 2 ) k ) = 2 + k 2, d a t b, G) = a t b h G = + ) 2 h a l b d a k, a l b = a k, a l b = k + 2 = k 2 + 2 h a k,a l b k i= h G a ki ) G + k ) + k i= = 2 + 2, a ki+l b) G )) 2 k k = 4 + 2 + k 4, where k ad l = 0,, 2,..., k. Thus DG) = { 2 + k 2, 4 + 2 + k 4 : k, k } ad a simple verificatio shows that DG) = 2τ).
278 Barzegar, Erfaia ad Farrokhi Now suppose that = 2 k m is eve, where k ad m is odd. The k + 2 k )) = 2 + k 2, k odd, d a k, G) = a k h a k h G = k 2 + 2 k 2)) = 2 + k, k eve. Sice a l b) G = {a l+2i b : 0 i 2 } we have d al b, G) = 2 + ) = 2 +. Also d a k, a l b, G) = 4 + k 4 +, 4 + k 2 +, k odd, k eve. Therefore { DD 2 ) = 2 + k, 2 + k 2 2, 4 + + k 3 2, 4 + + k 4 4 : k, k 2, k 3, k 4,, are eve ad, } are odd, k k 3 k 2 k 4 where k 3, k 4. We cosider the followig cases: ) If 2 + k = 4 + + k 3 2, the 2 = k 3+2 k ) ad k 3 +2 k ) < 0 if k >. Thus k = ad k 3 = 2. 2) If 2 + k = 4 + + k 4 4, the = k 4+4 k ) ad k 4 +2 k ) < 0 if k >. Thus k = ad k 4 =, which is a cotradictio. 3) If 2 + k 2 2 = 4 + + k 3 2, the 2 = k 3 + 2 k 2 ad k 3 + 2 k 2 < 0 if k 2 > 2. Thus k 2 2. Sice k 2 is odd we should have k 2 = 2, hece k 3 = 2. 4) If 2 + k 2 2 = 4 + + k 4 4, the = k 4 + 4 2k 2 ad k 4 + 4 2k 2 < 0 if k 2 > 2. Thus k 2 2, which is a cotradictio. 5) If 2 + k = 2 + k 2 2, the k 2 = 2k. 6) If 4 + + k 3 2 = 4 + + k 4 4, the k 4 = 2k 3. Now, by utilizig the cases )-6), the result follows. Corollary 4.2. DD 2 ) = 3 if ad oly if = p or 2p, where p is a prime. Defiitio 4.3. Let G ad G 2 be two groups ad H ad H 2 be subgroup of G ad G 2, respectively. Suppose that α is a isomorphism from G /ZG ) to G 2 /ZG 2 ) such that its restrictio to H /H ZG ) is a isomorphism from H /H ZG ) to H 2 /H 2 ZG 2 ) ad β is a isomorphism from [H, G ] to [H 2, G 2 ]. The the pair α, β) is called a 2
Relative commutativity degrees 279 relative isocliism from H, G ) to H 2, G 2 ) if the followig diagram is commutative: H H ZG ) G ZG ) α 2 H 2 H 2 ZG 2 ) G 2 ZG 2 ) where ad γ [H, G ] β γ 2 [H 2, G 2 ] γ h H ZG )), g ZG )) = [h, g ] γ 2 h 2 H 2 ZG 2 )), g 2 ZG 2 )) = [h 2, g 2 ] for each h H, h 2 H 2, g G ad g 2 G 2. If H = G ad H 2 = G 2, the we say that G ad G 2 are isocliic. As a immediate cosequet of the above defiitio we have the followig result. Lemma 4.4. If G ad G 2 are two isocliic groups, the DG ) = DG 2 ). Usig Lemma 4.4 ad Theorem 4. we obtai the followig results. Note that the geeralized quarerio groups Q 4 ad quasi-dihedral groups QD 2 3) are isocliic with the groups D 4 ad D 2, respectively. Corollary 4.5. If = 2 k m m odd), the DQ 4 ) = 2k +)τm). Corollary 4.6. If 3, the DQD 2 ) = 2 3. Ackowledgmets The secod author would like to thak Ferdowsi Uiversity of Mashhad for partial support of this research Grat No. MP90257ERF). Refereces [] A. Erfaia, R. Rezaei ad P. Lescot, O the relative commutativity degree of a subgroup of a fiite group, Comm. Algebra 35 2007), o. 2, 483 497. [2] A. Erfaia ad R. Rezaei, A ote o the relative commutativity degree of fiite groups, Submitted.
280 Barzegar, Erfaia ad Farrokhi [3] M. Farrokhi D. G., Fiite groups with two subgroup ormality degrees, Submitted. [4] W. H. Gustafso, What is the probability that two group elemets commute? Amer. Math. Mothly 80 973) 03 304. [5] P. Lescot, Isocliism classes ad commutativity degrees of fiite groups, J. Algebra 77 995), o. 3, 847 869. [6] F. Saeedi, M. Farrokhi D. G. ad S. H. Jafari, Subgroup ormality degrees of fiite groups I, Arch. Math. Basel) 96 20), o. 3, 25 224. R. Barzgar Departmet of Pure Mathematics, Ferdowsi Uiversity of Mashhad, P.O. Box 59, 9775 Mashhad, Ira Email: ro.gbps@gmail.com A. Erfaia Departmet of Pure Mathematics ad Ceter of Excellece i Aalysis o Algebraic Structures, Ferdowsi Uiversity of Mashhad, P.O. Box 59, 9775 Mashhad, Ira Email: erfaia@math.um.ac.ir M. Farrokhi D. G. Departmet of Pure Mathematics, Ferdowsi Uiversity of Mashhad, P.O. Box 59, 9775 Mashhad, Ira Email: m.farrokhi.d.g@gmail.com