Proceedings Mathematical Sciences z-classes in finite groups of conjugate type (n,1)

Proceedings Mathematical Sciences z-classes in finite groups of conjugate type (n,1) --Manuscript Draft-- Manuscript Number: Full Title: Article Type: Keywords: Corresponding Author: PMSC-D--00R1 z-classes in finite groups of conjugate type (n,1) Reseach Article conjugacy classes of centralizers, z-classes, p-groups, extraspecial groups. Indian Institute of Science Education and Research Mohali Mohali, INDIA Corresponding Author Secondary Information: Corresponding Author's Institution: Indian Institute of Science Education and Research Mohali Corresponding Author's Secondary Institution: First Author: First Author Secondary Information: Order of Authors: Shivam Arora Order of Authors Secondary Information: Funding Information: Abstract: Response to Reviewers: Two elements in a group $G$ are said to be \emph{$z$-equivalent} or to be in the same \emph{$z$-class} if their centralizers are conjugate in $G$. In a recent work Kulkarni, Kitture and Jadhav (see {J. Algebra Appl. doi:./s0}) proved that a non-abelian $p$-group $G$ can have at most $\frac{p^k-1}{p-1} +1$ number of $z$-classes, where $ G/Z(G) =p^k$. In this note, we characterize the $p$-groups of conjugate type $(n,1)$ attaining this maximal number. As a corollary, we characterize $p$-groups having prime order commutator subgroup and maximal number of $z$-classes. Dear Sir, We have revised the paper following the comments by the referee. We thank the referee for comments and suggestions. We would like to give only one clarification to the Comment number by the referee: we have made the bibliography as per the existing format in the Bibtex package of the AMS. All references given have been transported from the Bibtex file given on the AMS Mathscinet and appear as per the format set by the AMS Bibtex. The reference KKJ is yet to appear in the AMS or ZbMath. In this case the Bibtex reference has been collected from the journal website. Thank you again for reviewing your paper and kind comments! Best regards, Additional Information: Question Response Powered by Editorial Manager and ProduXion Manager from Aries Systems Corporation

Manuscript 1 z-classes IN FINITE GROUPS OF CONJUGATE TYPE (n,1) SHIVAM ARORA AND KRISHNENDU GONGOPADHYAY Abstract. Two elements in a group G are said to be z-equivalent or to be in the same z-class if their centralizers are conjugate in G. In a recent work Kulkarni, Kitture and Jadhav (see J. Algebra Appl. doi:./s0) proved that a non-abelian p-group G can have at most pk 1 + 1 number of z- classes, where G/Z(G) = p k. In this note, we characterize the p-groups of conjugate type (n, 1) attaining this maximal number. As a corollary, we characterize p-groups having prime order commutator subgroup and maximal number of z-classes. 1. Introduction Two elements in a group G are said to be z-equivalent or to be in the same z- class if their centralizers are conjugate in G. Since z-equivalence is an equivalence relation on G, it yields a partition of G into disjoint equivalence classes. It is coarser than the conjugacy relation. An infinite group may have infinitely many conjugacy classes, but in many groups the number of z-classes is finite. This finiteness of the z-classes gives a rough idea about dynamical types in a homogeneous space on which G acts, see [Kul0]. Following this motivation to use the z-classes to classify dynamical types of transformations, the z-classes of real hyperbolic isometries have been classified and counted by Gongopadhyay and Kulkarni [GK0]. Apart from geometric motivations, the z-classes are important objects in their own right. Characterizations of the z-classes in groups and explicit computations of their numbers is an interesting problem to study. A related problem is to compute the genus number in a linear algebraic group, see [Bos]. The z-classes play an important role in the study of characters of finite groups of Lie type, eg. see Carter [Car1]. Steinberg [Ste] proved that the number of z-classes is finite in a reductive algebraic group over a field of good characteristic. The z-classes in general linear groups and affine groups over arbitrary fields have been classified by Kulkarni [Kul0]. Gouraige [Gou0] obtained characterization of the z-classes in general linear groups over division rings. Singh [Sin0] has computed the z-classes of semisimple elements in the compact real form G. Gongopadhyay and Kulkarni [GK] have investigated the z-classes in orthogonal and symplectic groups over Date: June,. Mathematics Subject Classification. D (Primary); E (Secondary). Key words and phrases. conjugacy classes of centralizers, z-classes, p-groups, extraspecial groups. Gongopadhyay s research partially supported by SERB-DST grant SR/FTP/MS-00/. 1

S. ARORA AND K. GONGOPADHYAY fields of characteristic different from two. Gongopadhyay [Gon] has classified the z-classes in the isometry groups of the hyperbolic spaces and counted them for the quaternionic hyperbolic space. In a recent work Kulkarni, Kitture and Jadhav [KKJ] have investigated z-classes in finite p-groups. They proved that a non-abelian p-group has at least p + z- classes and characterized the groups that attain this lower bound. They also proved that a non-abelian p-group can have at most pk 1 + 1 number of z-classes, where G/Z(G) = p k. They proved the following theorem that gives necessary conditions for a p-group G to attain this maximal number of z-classes. Theorem KKJ. [KKJ, TheoremA] Let Gbe a non-abelian group with G/Z(G) = p k. If the number of z-classes in G is pk 1 + 1 then either G/Z(G) = C p C p or the following holds: (1) G has no abelian subgroup of index p. () G/Z(G) is an elementary abelian p-group. Further, a counter-example was given in [KKJ] to show that the above conditions (1) and () are not sufficient to attain this bound. It would be interesting to obtain sufficient conditions for a group G to attain the maximal number of z-classes and to characterize such groups. In this note, we characterize the groups of conjugate type (n, 1) attaining this bound. To state our result, we recall the following definition by Ito [Itô]. Definition. [Itô] Let G be a finite group. Let n 1,n,...,n r be integers, where n 1 > n >... > n r = 1, be all the numbers which are the indices of the centralizers of elements of G in G. The vector (n 1,n,...,n r ) is called conjugate type vector of G. A group with the conjugate type vector (n 1,n,...,n r ) is called a group of conjugate type (n 1,n,...,n r ), or simply, a group of type (n 1,n,...,n r ). In [Itô], Ito investigated the groups of type (n,1). Any such group is nilpotent and n = p a for some prime p. A group of type (p a,1) is the direct product of a p-group of the same type and of an abelian group. Thus, the study of groups of type (n,1) can be reduced to that of p-groups of type (p a,1). Ito showed that such a group G contains an abelian normal subgroup A such that G/A is of exponent p. Here we prove the following. Theorem 1.1. Let G be a non-abelian group of type (n,1) and [G : Z(G)] = p k. Then G has pk 1 +1 z-classes if and only if (1) G/Z(G) is elementary abelian, and () for all x G\Z(G), Z(C G (x)) = x,z(g). As an application of the above result we have the following. Corollary 1.. Let G be a non-abelian group with G/Z(G) = p k, (k ) and G = p. Then the number of z-classes in G is pk 1 +1 if and only if G is isoclinic to an extraspecial p-group.

z-classes IN FINITE GROUPS OF CONJUGATE TYPE (n, 1) We prove these results in Section. We note that Corollary 1. may be compared with Theorem..1 of Kitture [Kit, Section.]. We thank Kitture for letting us know about his result and many comments. We are also thankful to Silvio Dolfi for useful comments and suggestions on this work.. Preliminaries We recall a few basic facts that will be used in the proof of the above results. Let G be a group. We denote C G (x) to be the centralizer of an element x in G. Definition.1. Let G be a group. Two elements x,y G are said to be z-equivalent if their centralizers are conjugate, i.e. there exist g G such that, g 1 C G (x)g = C G (y). Let G be a group acting on a set X. For x X let G(x) denote the orbit of x and G x be the stabilizer subgroup of x: We have a partition of X as: G(x) = {y X : there exists g G and y = gx}. G x = {g G : gx = x}. X = x X G(x). Let y G(x) be some element, then for some g G we have y = gx and G y = gg x g 1. For subgroups A and B in G write A g B if they are conjugate. This gives us the following. Definition.. Let x,y X be any two element, they are said to be in same orbit class if G x g G y. Denote this equivalence relation as x 0 y and denote R(x) as the equivalence class of x. This gives another partition of X: X = x 0 y Let F x denotes the points fixed by G x i.e. R(x). F x = {y X : G y G x }. Let F x = {y X : G y = G x }. Define W x = N x /G x, where N x is the normalizer of the G x in G. Then we have the following theorem (cf. [Kul0]). Theorem.. [Kul0, Theorem.1] Let G act on a set X. Then the map φ : G/G x F x R(x), φ(gg x,y) = gy is well defined and induces a bijection, φ : (G/G x F x )/W x R(x).

S. ARORA AND K. GONGOPADHYAY When we take our space X = G, the group itself, then R(x) in above notation becomes the z-classes and the above theorem gives us the following equation, which is used to calculate the size of a z-class: for any x G we have z class of x = [G : N G (C G (x))]. F x, where C G (x) is the centralizer of x and F x = {y G : C G (y) = C G (x)}. The following theorem states that that number of z-classes is invariant under the isoclinism. For basic notions on isoclinism, see [Hal, KKJ]. Theorem.. [KKJ, Theorem.] Let G 1 and G be isoclinic groups, with an isoclinism (φ, ψ). Then the isoclinism (φ, ψ) induces a bijection between the z-classes in G 1 and G. Finally we note the following theorem by Hall [Hal] that is used in proof of Theorem 1.. Theorem.. Every group is isoclinic to a group whose center is contained in the commutator subgroup.. Proof of The Main Results It was proved in [KKJ, Lemma.1] that if G is a finite group with G/Z(G) = p k, then G is isoclinic to a finitep-group. Sincenumberof z-classes is invariant under isoclinism, hence it is enough to prove our results assuming that G is a p-group. Lemma.1. Let G be a non-abelian p-group with [G : Z(G)] = p k. If (1) G/Z(G) is elementary abelian, and () for all z G\Z(G), Z(C G (x)) = x,z(g), then G has pk 1 +1 z-classes. Proof. Let x G \ Z(G). Since G/Z(G) is abelian, hence C G (x) G, i.e. [G : N(C G (x)] = 1. Consider F x = {y G : C G (y) = C G (x)}. Bydefinition F x Z(C G (x))\z(g) = x,z(g) \Z(G). Obviously C G (x) = C G (xt) for t Z(G). Also since exponent of G/Z(G) is p, thus C G (x) = C G (x i ) for any i {1,,..,p 1}. Hence x,z(g) \Z(G) F x. Hence F x = Z(C G (x))\z(g) = x,z(g) \Z(G) = xz(g) x Z(G). This implies, F x = () Z(G). Therefore, z class of x = [G : N G (C G (x))]. F x = 1.() Z(G). Thus G\Z(G) contributes to a total pk 1 number of z-classes of G. Clearly, Z(G) is also a z-class. Hence, G has pk 1 +1 z-classes. Corollary.. Let G be a non-abelian p-group with G/Z(G) = p k. If (1) G has no abelian subgroup of order exceeding p Z(G). () G/Z(G) is an elementary abelian p-group.

Then G has pk 1 z-classes IN FINITE GROUPS OF CONJUGATE TYPE (n, 1) +1 number of z-classes. Proof. We claim C G (x) = x,z(g), and the rest follow from Lemma.1. Clearly x,z(g) C G (x). Let us assume y / x,z(g) and y C G (x). But then x,y,z(g) is an abelian subgroup and x,y,z(g) > p Z(G). Thus C G (x) = x,z(g)..1. Proof of Theorem 1.1. Proof. Let G be an non-abelian group of type (n,1) and G has pk 1 +1 z-classes, where [G : Z(G)] = p k. (1) follows from Theorem KKJ. We prove (). Let x G \ Z(G), then for any t Z(G), C G (x) = C G (xt). Also for any m relatively prime to order of x, we have Thus z-class of x contains, C G (x) = C G ( x ) = C G ( x m ) xz(g) x Z(G) x Z(G), and hence, for all x G\Z(G), the size of a z-class is at least () Z(G). Since Z(G) is also a z-class, hence it follows from the hypothesis that G\Z(G) constitute exactly pk 1 z-classes. Now, G \Z(G) = G Z(G). Hence, we must have for x G\Z(G), z class of x = () Z(G). Now, it is clear that for x G \ Z(G), x,z(g) Z(C G (x)). Now let y Z(C G (x)) such that y / x,z(g). Then a simple observation shows that C G (y) C G (x). But since G is of type (n,1) we have C G (x) = C G (y), thus we must have C G (x) = C G (y). This implies that y belongs to the z-class of x. Since y / x,z(g), z class of x > () Z(G), which is a contradiction. Thus our assumption is incorrect and Z(C G (x)) = x,z(g) for all x G\Z(G). The converse part follows from Lemma.1. This completes the proof... Proof of Corollary 1.. Proof. Without loss of generality, using Theorem., we assume up to isoclinism that, G is a group such that Z(G) G. Let G has pk 1 + 1 z-classes. Since G = p and in a non-abelian p-group Z(G) > 1, it follows that G = Z(G); hence, G = Z(G) = p. Therefore G and G/Z(G) have the same exponent. Thus G is an extra-special p-group. Conversely, let G be an extraspecial p-group. So, Z(G) = G = p. For x G\Z(G), define the map: φ x : G G g [x,g] As G/Z(G) is elementary abelian, φ x is a homomorphism with ker(φ x ) = C G (x) and since G = p, the map is also surjective. Hence we get G/C G (x) = G

S. ARORA AND K. GONGOPADHYAY ThuseachC G (x)isamaximalsubgroupofindexping. Thisshowsthatextraspecial p-groups are of type (n,1). Now, we claim that Z(C G (x)) = x,z(g). This follows from the fact that G/Z(G) can be equipped with an alternating non-degenerate form induced by the commutator, see [Cra0, Theorem.] or [Wil0, p. ]. The claim follows from [Cra0, Lemma.] noting that Z(C G (x))/z(g) is the radical of co-dimension one subspace C G (x)/z(g). The result now follows from Theorem 1.1. References [Bos] Anirban Bose. On the genus number of algebraic groups. J. Ramanujan Math. Soc., ():,. [Car1] R. W. Carter. Centralizers of semisimple elements in the finite classical groups. Proc. London Math. Soc. (), (1):1, 1. [Cra0] David A. Craven. The theory of p-groups. (available at the author s homeage). 0. [GK0] and Ravi S. Kulkarni. z-classes of isometries of the hyperbolic space. Conform. Geom. Dyn., :1, 0. [GK] and Ravi S. Kulkarni. The Z-classes of isometries. J. Indian Math. Soc. (N.S.), 1(-):,. [Gon]. The z-classes of quaternionic hyperbolic isometries. J. Group Theory, ():,. [Gou0] Rony Gouraige. Z-classes in central simple algebras. ProQuest LLC, Ann Arbor, MI, 0. Thesis (Ph.D.) City University of New York. [Hal] P. Hall. The classification of prime-power groups. J. Reine Angew. Math., :1 1,. [Itô] Noboru Itô. On finite groups with given conjugate types. I. Nagoya Math. J., :,. [Kit] Rahul D. Kitture. z-classes in finite p-groups.. Thesis (Ph.D.) University of Pune. [KKJ] Ravindra Kulkarni, Rahul Dattatraya Kitture, and Vikas S. Jadhav. z-classes in groups. Journal of Algebra and Its Applications, 0(0):1, doi:./s0. [Kul0] Ravi S. Kulkarni. Dynamical types and conjugacy classes of centralizers in groups. J. Ramanujan Math. Soc., (1):, 0. [Kul0] Ravi S. Kulkarni. Dynamics of linear and affine maps. Asian J. Math., ():1, 0. [Sin0] AnupamSingh.Conjugacyclasses ofcentralizers ing.j. Ramanujan Math. Soc., ():, 0. [Ste] Robert Steinberg. Conjugacy classes in algebraic groups. Lecture Notes in Mathematics, Vol.. Springer-Verlag, Berlin-New York,. Notes by Vinay V. Deodhar. [Wil0] Robert A. Wilson. The finite simple groups, volume 1 of Graduate Texts in Mathematics. Springer-Verlag London, Ltd., London, 0. Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City, Sector 1, S.A.S. Nagar, Punjab 0, India E-mail address: shivam.iiserm@gmail.com Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City, Sector 1, S.A.S. Nagar, Punjab 0, India E-mail address: krishnendug@gmail.com, krishnendu@iisermohali.ac.in