Counting centralizers and z-classes of some F-groups

arXiv:2011.14071v2 [math.GR] 3 Dec 2020 COUNTING CENTRALIZERS AND z-CLASSES OF SOME F-GROUPS SEKHAR JYOTI BAISHYA Abstract. A finite group G is called an F-group if for every x, y ∈ G \ Z(G), C(x) ≤ C(y) implies that C(x) = C(y). On the otherhand, two elements of a group are said to be z-equivalent or in the same z-class if their centralizers are conjugate in the group. In this paper, for a finite group, we give necessary and sufficient conditions for the number of centralizers/ z-classes to be equal to the index of its center. We also give a necessary and sufficient condition for the number of z-classes of a finite F-group to attain its maximal number (which extends an earlier result). Among other results, we have computed the number of element centralizers and z-classes of some finite groups and extend some previous results. 1. Introduction In 1953, Ito [21] introduced the notion of the class of F-groups, consisting of finite groups G in which for every x, y ∈ G \ Z(G), C(x) ≤ C(y) implies that C(x) = C(y), where C(x) and C(y) are centralizers of x and y respectively. Since then the influence of the element centralizers on the structure of groups has been studied extensively. An interesting subclass of F-groups is the class of I-groups, consisting of groups in which all centralizers of non-central elements are of same order. Ito in [21] proved that I-groups are nilpotent and direct product of an abelian group and a group of prime power order. Later on in 2002, Ishikawa [19] proved that I-groups are of class at most 3. In 1971, Rebmann [33] investigated and classified F-groups. In 2007, R. S. Kulkarni [26,27] introduced the notion of z-classes in a group. Two elements of a group are said to be z-equivalent or in the same z-class if their centralizers are conjugate in the group. z-equivalence is an equivalence relation which is weaker that conjugacy relation. An infinite group generaly contains infinitely many conjugacy classes, but may have finitely many z-classes. In [27] the author observed the influence of the z-classes in the groups of automorphisms of classical geometries and apart from other results he concludes that this finiteness of z-classes can be related to the idea of finiteness of dynamical types of transformation to the geometry. It may be mentioned here that apart from the geometric motivation, finding 2010 Mathematics Subject Classification. 20D60, 20D99. Key words and phrases. Finite group, Centralizer, Partition of a group, z-class. 1 2 S. J. BAISHYA z-classes of a group itself is of independent interest as a pure combinatoral problem. More information on this and related concepts may be found in [14–16, 23, 29]. In a recent work, the authors in [28], investigated z-classes in finite p-groups. Among other results, they proved that a non-abelian p-group can have at most pk −1 G + 1 number of z-classes, where | Z(G) |= pk and gave the necessary condition to p−1 attain the maximal number which is not sufficient. Recently, the authors in [6], gave a necessary and sufficient condition for a finite p-group of conjugate type (n, 1) to attain this maximal number. In this paper, apart from other results, we extend this result and give a necessary and sufficient condition for a finite F -group to attain this maximal number. For a finite group, we give necessary and sufficient conditions for the number of centralizers/ z-classes to be equal to the index of its center. As a consequence, we confirm a Conjecture in [4], namely, if G is a finite group such that the number of centralizers is equal to the index of its center, then G is an F-group. Among other results, we have computed the number of element centralizers and z-classes of some groups and improve some earlier results. It may be mentioned here that characterization of groups in terms of the number of element centralizers have been considered by many researchers (see for example [37] for finite groups and [39] for infinite groups). Throughout this paper, G is a group with center Z(G), commutator subgroup G′ and the set of element centralizers Cent(G). We write Z(x) to denote the center of the proper centralizer C(x) and ‘z-class’ to denote the set of z-classes in G. 2. Preliminaries We begin with some Remarks which will be used in the sequel. Remark 2.1. (See [36, Pp. 571–572]) A collection Π of non-trivial subgroups of a group G is called a partition if every non-trivial element of G belongs to a unique subgroup in Π. If | Π |= 1, the partition is said to be trivial. The subgroups in Π are called components of Π. Following Miller, any abelian group having a non-trivial partition is an elementary abelian p-group of order ≥ p2 . Let S be a subgroup of G. A set Π = {H1 , H2 , . . . , Hn } of subgroups of G is said to be a strict S-partition of G if S ≤ Hi (i = 1, 2, . . . , n) and every element of G \ S belongs to one and only one subgroup Hi (i = 1, 2, . . . , n). For more information about partition see [34]. Given a group G, let A = {C(x) | x ∈ G \ Z(G)} and B = {Z(x) | x ∈ G \ Z(G)}. A and B are partially ordered sets with respect to inclusion and they have the same length. The length of A (and of B) is called the rank of G. A group G has rank 1 if and only if B is a strict Z(G) partition. Recall that a finite group G is called an F-group if for every x, y ∈ G \ Z(G), C(x) ≤ C(y) implies that C(x) = C(y). Following [12, Lemma 2.6], a finite nonabelian group G is an F-group if and only if B is a strict Z(G) partition. Hence being a group of rank 1 is equivalent to being an F-group. COUNTING CENTRALIZERS AND z-CLASSES OF SOME F-GROUPS 3 Remark 2.2. Given a group G, two elements x, y ∈ G are said to be z-equivalent or in the same z-class if their centralizers are conjugate in G, i.e., if C(x) = gC(y)g −1 for some g ∈ G. It is well known that “being z-equivalent” is an equivalenve relation on G. Following [27, Theorem 2.1], the order of the z-class of x, if finite, is given by | z − class of x |=| G : NG (C(x)) | . | Fx′ |, where Fx′ := {y ∈ G | C(x) = C(y)}. From this it is easy to see that the number of z-classes in G i.e., | z − class |=| Cent(G) | if and only if C(x) ✁ G for all x ∈ G. The following Theorems will be used to obtain some of our results. For basic notions of isoclinism, see [17, 28]. Theorem 2.3. (p.135 [17]) Every group is isoclinic to a group whose center is contained in the commutator subgroup. Theorem 2.4. (Theorem A [22], Lemma 3.2 [39]) Any two isoclinic groups have the same number of centralizers. Theorem 2.5. (Theorem 11 [22], Theorem 3.3 [39]) The representatives of the families of isoclinic groups with n-centralizers (n 6= 2, 3) can be chosen to be finite group. Theorem 2.6. (Lemma 4, p. 303 [11]) Let G be a finite group with an abelian normal subgroup of prime index p. Then | G |= p. | Z(G) | . | G′ |. 3. The main results In this section, we prove the main results of the paper. Let D8 be the dihedral D8 |. The following group of order 8. It is easy to verify that | Cent(D8 ) |=| Z(D 8) G |, where G is result gives a necessary and sufficient condition for | Cent(G) |=| Z(G) a finite group. Recall that Z(x) denotes the center of the proper centralizer C(x). Proposition 3.1. Let G be a finite group. Then | Cent(G) |=| for all x ∈ G \ Z(G). G Z(G) | iff | Z(x) Z(G) |= 2 G |= n and C(xi ), 1 ≤ i ≤ n − 1 be the proper Proof. Let | Cent(G) |=| Z(G) centralizers of G. Suppose xi Z(G) = xj Z(G) for some 1 ≤ i, j ≤ n − 1; i 6= j. Then xi z1 = xj z2 for some z1 , z2 ∈ Z(G) and consequently, C(xi ) = C(xi z1 ) = C(xj z2 ) = C(xj ), which is a contradiction. Hence G = Z(G) ⊔ x1 Z(G) ⊔ x2 Z(G) ⊔ · · · ⊔ xn−1 Z(G). Now, for some i, 1 ≤ i ≤ n − 1 suppose Z(xi ) = Z(G) ⊔ xi Z(G) ⊔ X, where X = ⊔xj Z(G) for some 1 ≤ j ≤ n − 1, i 6= j. Then C(xi ) = C(xl ) for some Z(x) 1 ≤ l ≤ n − 1, i 6= l, which is a contradiction. Therefore | Z(G) |= 2 for all x ∈ G \ Z(G). 4 S. J. BAISHYA Z(x) G |= 2 for all x ∈ G \ Z(G). Then | Cent(G) |=| Z(G) |, Conversely, suppose | Z(G) noting that in the present scenario, for any x ∈ G \ Z(G), Z(x) will contain exactly one right coset of Z(G) other than Z(G).  The authors in [28, Proposition 2.4] proved that if G is a group in which Z(G) has finite index, then the number of z-classes is at most the index [G : Z(G)]. In the following result, for a finite group we give a necessary and sufficient condition for | z − class | to attain its upper bound. We also confirm a Conjecture in [4], G namely if G is a finite group such that | Cent(G) |=| Z(G) |, then G is an F-group. Theorem 3.2. Let G be a finite group. G (a) If | Cent(G) |=| Z(G) |, then G is an F-group. ( [4, Conjecture 2.5]). G G is elementary abelian 2-group. ( [4, Theo(b) If | Cent(G) |=| Z(G) |, then Z(G) rem 2.1]) Z(x) G | iff | Z(G) |= 2 for all x ∈ G \ Z(G). (c) | z − class |=| Z(G) G |, then G is an F-group. (d) If | z − class |=| Z(G) Z(x) Proof. a) By Proposition 3.1, we have | Z(G) |= 2 for all x ∈ G \ Z(G) and hence Z(x) ∩ Z(y) = Z(G) for any x, y ∈ G \ Z(G) with C(x) 6= C(y). Therefore in view of Remark 2.1, G is an F-group. b) By Proposition 3.1, we have | Z(x) Z(G) |= 2 for all x ∈ G \ Z(G). Therefore Z(x) Π = { Z(G) | x ∈ G \ Z(G)} is a partition of G Z(G) and hence the result follows. G G c) Suppose | z − class |=| Z(G) |. By [28, Proposition 2.5], Z(G) is abelian and G hence using Remark 2.2, | Cent(G) |=| z − class |=| Z(G) |. Now, the result follows from Proposition 3.1. Z(x) |= 2 for all x ∈ G\Z(G). Then in view of Proposition Converesely, suppose | Z(G) G |. 3.1, Corollary 3.2 and Remark 2.2, | z − class |=| Cent(G) |=| Z(G) Z(x) d) From(c), we have | Z(G) |= 2 for all x ∈ G\Z(G) and hence Z(x)∩Z(y) = Z(G) for any x, y ∈ G \ Z(G) with C(x) 6= C(y). Therefore in view of Remark 2.1, G is an F-group.  Recall that a p-group (p a prime) is said to be special if its center and commutator subgroup coincide and are elementary abelian. Furthermore, a group G is extraspecial if G is a special p-group and | G′ |=| Z(G) |= p. We now give the following result concerning the upper and lower bounds of | Cent(G) |. For lower bounds of | Cent(G) | one may also see [38, Corollary 2.2]. Theorem 3.3. Let G be a finite group and p be the smallest prime divisor of its order. Then COUNTING CENTRALIZERS AND z-CLASSES OF SOME F-GROUPS 5 G ∼ (a) p + 2 ≤| Cent(G) |; with equality if and only if Z(G) = Cp × Cp ; equivalently, if and only if G is isoclinic to an extraspecial group of order p3 . Z(x) G |; with equality if and only if | Z(G) |= 2 for all x ∈ (b) | Cent(G) |≤| Z(G) G \ Z(G); equivalently, G is isoclinic to a special 2-group. Proof. a) Using arguments similar to [7, Lemma 2.7], we have p + 2 ≤| Cent(G) | G ∼ with equality if and only if Z(G) = Cp × Cp . Next, suppose G is isoclinic to an extraspecial group of order p3 . Then by Theorem 2.4 and [10, Theorem 5], we have p + 2 =| Cent(G) |. G ∼ Conversely, Suppose p + 2 =| Cent(G) |. Then we have Z(G) = Cp × Cp . In the present scenario, by Theorem 2.3 and Theorem 2.5, G is isoclinic to a finite H ∼ group H of order pn with Z(H) ⊆ H ′ . Moreover, since Z(H) = Cp × Cp , therefore ′ Z(H) = H and H has an abelian subgroup of index p. Hence using Theorem 2.6, we have pn = p. | Z(H) | . | H ′ | and consequently, H is an extraspecial group of order p3 . b) Since C(xz1 ) = C(xz2 ) for all xz1 , xz2 ∈ xZ(G), therefore | Cent(G) |≤| G Z(G) |; Z(x) Z(G) with equality if and only if | |= 2 for all x ∈ G \ Z(G) by Proposition 3.1. G G |. Then in view of Theorem 3.2, Z(G) For the last part, suppose | Cent(G) |=| Z(G) is elementary abelian 2-group. Therefore G = A × H, where A is an abelian group H and H is the sylow 2-subgroup of G with Z(H) elementary abelian. Consequently, G is isoclinic to H. In the present scenario, by Theorem 2.3 and Theorem 2.5, H1 H is isoclinic to a finite group H1 of order pn with Z(H1 ) ⊆ H1′ . Since Z(H 1) H1 have same is abelian, therefore Z(H1 ) = H1′ . It now follows that H1′ and Z(H 1) exponent (by [2, Lemma 9, p. 77]). In the present scenario, we have H1 = Z(H1 ) is elementary abelian 2-group and hence H1 is a special 2-group isoclinic to G.  We now give our first counting formula for number of distinct centralizers: Proposition 3.4. Let G be a finite F-group such that | for all x ∈ G \ Z(G), then | Cent(G) |= pk −1 pm −1 G Z(G) |= pk . If | Z(x) Z(G) |= pm + 1. Proof. Since G is an F-group, therefore by Remark 2.1 we have Z(x) ∩ Z(y) = Z(G) for any x, y ∈ G\Z(G) with C(x) 6= C(y). Hence the result follows, noting that each Z(x) contains exactly (pm − 1) distinct right cosets of Z(G) other than Z(G).  The following result generalizes [3, Theorem 3.5]. Theorem 3.5. Let G be a finite group and p a prime. Then | x ∈ G \ Z(G) if and only if (a) G is an F-group. G (b) Z(G) is of exponent p. Z(x) Z(G) |= p for all 6 S. J. BAISHYA (c) | Cent(G) |= pk −1 p−1 Z(x) Z(G) Z(x) { Z(G) + 1, where | G Z(G) |= pk . Proof. Suppose | |= p for all x ∈ G \ Z(G). a) Clearly, Π = G is an F-group. | x ∈ G \ Z(G)} is a partition of G . Z(G) b) It is clear from the proof of (a) that every element of Hence by Remark 2.1, G Z(G) is of order ≤ p. c) Clearly, for any x ∈ G \ Z(G), Z(x) contains exactly (p − 1) distinct right cosets of Z(G) different from Z(G). And hence the result follows. k −1 G |= pk and | Cent(G) |= pp−1 + 1(= l). Conversely, suppose G is an F-group, | Z(G) Let C(xi ), 1 ≤ i ≤ l − 1 be the proper centralizers of G. Since G is an F-group, G i) | 1 ≤ i ≤ l − 1} is a partition of Z(G) . In therefore by Remark 2.1, Π = { Z(x Z(G) the present scenario, we have pk =| Consequently, | Z(x) Z(G) Z(x1 ) Z(G) |+| Z(x2 ) Z(G) | +···+ | |= p for all x ∈ G \ Z(G). Z(xl−1 ) Z(G) k −1 | − pp−1 + 1.  As an immediate corollary, we have the following result ( [3, Theorem 3.5]). Recall that a finite group G is said to be a CA-group if centralizer of every noncentral element of G is abelian. Corollary 3.6. Let G be a finite F-group such that | CA-group, then | Cent(G) |= p3 + p2 + p + 2. G Z(G) |= p4 . If G is not a Proof. It follows from Theorem 3.5, noting that in the present scenario, we have Z(x) |= p for all x ∈ G \ Z(G).  | Z(G) Following Ito [21], a finite group G is said to be of conjugate type (n, 1) if every proper centralizer of G is of index n. He proved that a group of conjugate type (n, 1) is nilpotent and n = pa for some prime p. Moreover, he also proved that a group of conjugate type (pa , 1) is a direct product of a p-group of the same type and an abelian group. The author in [20] classified finite p-groups of conjugate type (p, 1) and (p2 , 1) upto isoclinism. In the following result, we calculate the number of element centralizers and z-classes of a finite group of conjugate type (p, 1). Given a group G, nacent(G) denotes the set of non-abelian centralizers of G. For more information about nacent(G) see [5, 24]. G Proposition 3.7. Let G be a finite group such that | C(x) |= p for all x ∈ G \ Z(G). Then G (a) Z(G) is elementary abelian p-group of order pk for some k. k −1 (b) | Cent(G) |=|z−class |= pp−1 + 1. G ∼ (c) | nacent(G) |= 1 iff Z(G) = Cp × Cp . COUNTING CENTRALIZERS AND z-CLASSES OF SOME F-GROUPS (d) | nacent(G) |=| Cent(G) | iff G Z(G) 7 is of order > p2 . Proof. a) In view of Ito [21], G = A × P , where A is an abelian group and P is a G is elementary p-group of conjugate type (p, 1). Therefore using Remark 2.1, Z(G) abelian p-group of order pk for some k. b) The result follows from Theorem 3.5 and Remark 2.2, noting that in the present Z(x) |= p for all x ∈ G \ Z(G). scenario, by [32, Proposition 1], we have | Z(G) G ∼ c) If Z(G) = Cp × Cp , then G is a CA-group and hence | nacent(G) |= 1. Conversely, suppose | nacent(G) |= 1. Then view of [32, Proposition 1], we have G ∼ C × Cp . Z(G) = p d) In view of [32, Proposition 1], we have | the result follows. Z(x) Z(G) |= p for all x ∈ G \ Z(G). Hence  We now compute the number of distinct centralizers and z-classes of an extraspecial p-group. It is well known that for any prime p and every positive integer a, there exists, upto isomorphism, exactly two extraspecial groups of order p2a+1 . Proposition 3.8. Let G be an extraspecial p-group of order p2a+1 for some prime 2a −1 p. Then | Cent(G) |=| z − class |= pp−1 + 1. Proof. Using [31, Pp. 5], we have | C(x) |= p2a for all x ∈ G \ Z(G). Now the result follows from Proposition 3.7, noting that here we have | Z(G) |= p.  Our next result concerns about finite groups of conjugate type (p2 , 1). G Proposition 3.9. Let G be a finite group such that | C(x) |= p2 for all x ∈ G\Z(G). Then one of the following assertions hold: G is elementary abelian p-group. (a) Z(G) G (b) Z(G) is non-abelian of order p3 (p odd) and of exponent p; and | Cent(G) |= p2 + p + 2. G G is abelian, then by Remark 2.1, Z(G) is elementary abelian. Next, Z(G) Z(x) G suppose Z(G) is non-abelian. In view of [32, Proposition 2], Z(G) = p for all x ∈ G G\Z(G). In the present scenario, if p = 2, then | Cent(G) |=| Z(G) | and hence using G is elementary abelian, which is a contradiction. Therefore p is Corollary 3.2, Z(G) 5 Proof. If odd and consequently, using [20, Theorem 4.2], G is isoclinic to a group of order p G with center of order p2 . Hence Z(G) is non-abelian of order p3 . Since G is an F-group, Z(x) G therefore by Remark 2.1, Π = { Z(G) | x ∈ G \ Z(G)} is a partition of Z(G) and hence G 2 is of exponent p. Moreover, by Theorem 3.4 we have | Cent(G) |= p +p+2.  Z(G) 8 S. J. BAISHYA It may be mentioned here that for the group G:= Small group (64, 73) in [40], G ∼ G we have Z(G) |= 4 for all x ∈ G \ Z(G). = C2 × C2 × C2 and | C(x) The following Proposition generalizes [3, Theorem 3.8]. Z(x) G Proposition 3.10. Let G be a finite F-group such that | Z(G) |= pk . If | Z(G) |≤ p2 for all x ∈ G \ Z(G), then | Cent(G) |= pk−1 + pk−2 + · · · + p + 2 − vp, where v is Z(x) = p2 . the number of centralizers for which Z(G) Proof. Since G is a finite F-group, therefore by Remark 2.1, Z(x) ∩ Z(y) = Z(G) for all x, y ∈ G \ Z(G) with C(x) 6= C(y). Let v be the number of centralizers for which Z(x) = p2 . Then the centers of these v number of centralizers will contain exactly Z(G) v(p2 − 1) distinct right cosets of Z(G) different from Z(G). On the other-hand the center of each of the remaining proper centralizers will contain exactly (p − 1) distinct right cosets of Z(G) other than Z(G). Consequently, | Cent(G) |= (pk − 1) − v(p2 − 1) + v + 1 = pk−1 + pk−2 + · · · + p + 2 − vp. p−1  As an immediate corollary we obtain the following result for finite groups of conjugate type (p2 , 1). As we have already mentioned, Ishikawa [19] proved that I-groups are of class at most 3. For finite groups of class 3 having conjugate type (p2 , 1), we obtained in Proposition 3.9 that | Cent(G) |= p2 + p + 2. G |= p2 for all x ∈ G \ Z(G). Corollary 3.11. Let G be a finite group such that | C(x) G |= pk and v is the Then | Cent(G) |= pk−1 + pk−2 + · · · + p + 2 − vp, where | Z(G) number of centralizers for which Z(x) Z(G) = p2 . Proof. It follows from Proposition 3.10, noting that in the present scenario, in view Z(x) |≤ p2 for all x ∈ G \ Z(G).  of [32, Proposition 1], we have | Z(G) We now prove the following result which improves [7, Theorem 3.3]. A group G is said to be n-centralizer if | Cent(G) |= n. Proposition 3.12. Let G be a finite n(= p2 +2)-centralizer group. Then | G ∼ for all x ∈ G \ Z(G) iff Z(G) = Cp × Cp × Cp × Cp and G is an F-group. G C(x) |= p2 G Proof. Suppose | C(x) |= p2 for all x ∈ G \ Z(G). Then G is an F-group. In view of G is elementary abelian. Let Xi = C(xi ), 1 ≤ i ≤ n − 1 where Proposition 3.9, Z(G) n−1 n−1 P xi ∈ G \ Z(G). We have G = ∪ Xi and | G |= | Xi |. Therefore by [13, Cohn’s i=1 i=1 Theorem], we have G = X1 X2 and X1 ∩X2 = Z(G). Hence G Z(G) ∼ = Cp ×Cp ×Cp ×Cp . COUNTING CENTRALIZERS AND z-CLASSES OF SOME F-GROUPS 9 G ∼ Conversely, suppose Z(G) = Cp × Cp × Cp × Cp and G is a finite F-group. In view G |= p for some x ∈ G \ Z(G). of Corollary 3.6, G is a CA-group. Now, suppose | C(x) Then in view of Theorem 2.6 and [9, Theorem 2.3], | Cent(G) |6= p2 + 2. Therefore Z(x) |≤ p2 for all x ∈ G \ Z(G) and consequently, using Corollary 3.11 we have | Z(G) G |= p2 for all x ∈ G \ Z(G).  | C(x) Remark 3.13. Let G be a finite group and let H be a subgroup of G. Then (G, H) is called a Camina pair if x is conjugate in G to xz for all x ∈ G \ H. A finite group G is called a Camina group if (G, G′ ) is a Camina pair. Recall that a p-group G is semi-extraspecial, if G satisfies the property for every G maximal subgroup N of Z(G) that N is an extraspecial group. It is known that every semi-extraspecial group is a special group ( [31, Pp.5]) and for such groups | C(x) | is equal to the index [G : G′ ] for all x ∈ G \ G′ ( [31, Theorem 5.5]). Following [31, Theorem 5.2], a group G is a semi-extraspecial p-group for some prime p if and only if G is a Camina group of nilpotent p class 2. A group G is said ′ to be ultraspecial if G is semi-extraspecial and | G |= | G : G′ |. It is known that all of the ultraspecial groups of order p6 are isoclinic ( [31, Pp. 10]). As an immediate application to the above Proposition, we have the following result: G Proposition 3.14. Let G be a finite n(= p2 +2)-centralizer group. Then | C(x) |= p2 for all x ∈ G \ Z(G) if and only if G is isoclinic to an ultraspecial group of order p6 . G is abelian. Therefore G is nilpotent of class Proof. In view of Proposition 3.9, Z(G) 2 and hence by [20, Theorem 4.1], Proposition 3.12 and Remark 3.13 we have the result.  In [8, Theorem 3.4], the author proved that if G is a finite 6-centralizer group, G ∼ then Z(G) = D8 , A4 , C2 × C2 × C2 or C2 × C2 × C2 × C2 . In this connection, we have the following result: G Proposition 3.15. Let G be a finite group such that | Z(G) |= 16. Then G is 6-centralizer if and only if G is isoclinic to an ultraspecial group of order 64. Proof. Suppose | Cent(G) |= 6. Using [1, Proposition 2.5 (a)], G is a CA-group. G |= 2 for some x ∈ G \ Z(G). Then in view of Theorem Now, suppose | C(x) 2.6 and [9, Theorem 2.3], | Cent(G) |6= 6. Consequently, by Proposition 3.10, G we have | C(x) |= 4 for all x ∈ G \ Z(G). Therefore by Proposition 3.14, G is isoclinic to an ultraspecial group of order 64. Conversely, suppose G is isoclinic to G an ultraspecial group of order 64. Then by Remark 3.13, | G′ |= 4 and | C(x) |= 4 G for all x ∈ G \ Z(G). Since we have | Z(G) |= 16, therefore by Proposition 3.4, | Cent(G) |= 6.  10 S. J. BAISHYA We now compute the number of centralizers of a finite group with maximal centralizers (maximal among the subgroups). It may be mentioned here that Kosvintsev in 1973 [25] studied and characterised these groups. He proved that in a finite nilpotent group every centralizer is maximal if and only if G is of the conjugate type (p, 1). Moreover, he proved that in a finite non-nilpotent group G the centralizer of every non-central is a maximal subgroup if and only if G = MZ(G), where M is a biprimary subgroup in G that is a Miller-Moreno group. Recently, in 2020 the authors [35] studied these groups. It seems that the authors are unaware of the paper of Kosvintsev [25]. However they have given a characterization of such G non-nilpotent group G in terms of Z(G) by proving that if G is a non-nilpotent Q G P G is either abelian or Z(G) = Z(G) ⋊ Z(G) is a minimal group of such type, then Z(G) non-abelian group (Miller and Moreno analyzed minimal non-abelian groups in [30]) Q P with | Z(G) |= pa and | Z(G) |= q, where p and q are primes. In this connection, we G cannot be abelian. notice that there is a minor error in the result. In this case Z(G) Proposition 3.16. Let G be a finite group in which the centralizer of any noncentral element of G is maximal (maximal among all subgroups). G (a) If G is nilpotent, then | Z(G) |= pk for some k ∈ N and | Cent(G) |=|z−class |= pk −1 + 1. Moreover, | nacent(G) |= 1 iff G ∼ = Cp × Cp and | nacent(G) |=| p−1 Z(G) G is of order > p2 . Cent(G) | iff Z(G) G (b) If G is non-nilpotent, then | Cent(G) |=| Cent( Z(G) ) |= pa + 2, where pa is G . Moreover, G is a CA-group. the order of the p-Sylow subgroup of Z(G) G |= p (p a Proof. a) Since G is nilpotent, therefore in view of [25], we have | C(x) prime) for all x ∈ G \ Z(G). Now, the result follows using Proposition 3.7. Q G P b) In view of [35, Theorem A], we have Z(G) = Z(G) ⋊ Z(G) is a minimal nonQ P a abelian group with | Z(G) |= p and | Z(G) |= q, where p and q are primes. Moreover, G by [18, Aufgaben III, 5.14], Z(G) has trivial center. In the present scenario, we have C(x) Z(G) G = C(xZ(G)) for any x ∈ G \ Z(G). Hence | Cent(G) |=| Cent( Z(G) ) |= pa + 2 and G is a CA-group, noting that in the present scenario G is an F-group.  The authors in [6, Main Theorem 1.1], gave a necessary and sufficient condition for a finite p-group of type (n, 1) to attain the maximal number of z-classes. In the following Theorem we extend this result as follows: Theorem 3.17. Let G be a finite F-group with | Then G has (a) G Z(G) pk −1 p−1 + 1 z-classes if and only if is elementary abelian and G Z(G) |= pk , where p is a prime. COUNTING CENTRALIZERS AND z-CLASSES OF SOME F-GROUPS 11 (b) For all x ∈ G \ Z(G), Z(x) = hx, Z(G)i. k −1 G |= pk and G has pp−1 + 1 z-classes. Proof. Let G be a finite F-group such that | Z(G) By [28, Lemma 3.1], G is isoclinic to a finite p-group H and by [28, Theorem 2.2], k −1 H + 1 z-classes. In the present scenario, by [28, Theorem 3.13], Z(H) is H has pp−1 G elementary abelian and consequently, Z(G) is elementary abelian. Hence C(x) ✁ G k −1 for all x ∈ G and therefore, by Remark 2.2, | Cent(G) |= pp−1 + 1. Now, the result follows from Theorem 3.5. Conversely, suppose (a) and (b) holds. Then C(x) ✁ G for all x ∈ G and consek −1 quently, by Remark 2.2 and Theorem 3.5 we have | z − class |= pp−1 + 1.  We conclude the paper with the following result: Proposition 3.18. Let G be an finite group such that | G′ |= p (p a prime) and G′ ⊆ Z(G). Then G is isoclinic to an extraspecial p-group of order p2a+1 and 2a −1 | Cent(G) |=| z − class |= pp−1 + 1. Proof. Since G′ ⊆ Z(G) therefore G = A × H, where A is an abelian subgroup and H is the sylow p-subgroup of G with | H ′ |= p. Consequently, G is isoclinic to H. In the present scenario, by Theorem 2.3 and Theorem 2.5, H is isoclinic to a finite p-group H1 with Z(H1 ) ⊆ H1′ . Therefore we have | Z(H1 ) |=| H1′ |= p. It H1 have same exponent (by [2, Lemma 9, p. 77]). Thus now follows that H1′ and Z(H 1) H1 is an extraspecial p-group of order p2a+1 and G is isoclinic to H1 . Moreover, by 2a −1 Proposition 3.8 and Theorem 2.4 we have | Cent(G) |=| z − class |= pp−1 + 1.  Acknowledgment I would like to thank Prof. Mohammad Zarrin for his valuable suggestions and comments on the earlier draft of the paper. References [1] A. Abdollahi, S. M. J. Amiri and A. M. 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Email address: [email protected]