Groups having complete bipartite divisor graphs for their conjugacy class sizes

Groups having complete bipartite divisor graphs for their conjugacy class sizes

ROGHAYEHHAFEZIEH(*) - PABLOSPIGA(**)

ABSTRACT- Given a finite groupG, thebipartite divisor graphfor its conjugacy class sizes is the bipartite graph with bipartition consisting of the set of conjugacy class sizes ofGnZ(G)(whereZ(G)denotes the centre ofG) and the set of prime numbers that divide these conjugacy class sizes, and withfp;ngbeing an edge if gcd(p;n)6ˆ1.

In this paper we construct infinitely many groups whose bipartite divisor graph for their conjugacy class sizes is the complete bipartite graphK2;5, giving a solution to a question of Taeri [15].

MATHEMATICSSUBJECTCLASSIFICATION(2010). Primary 00Z99; secondary 99A00.

KEYWORDS. Bipartite divisor graph, conjugacy class size, extra-special group.

1. Introduction

Given a finite groupG, there are several sets of invariants that convey nontrivial information about the structure ofG. Some classical examples include the set consisting of the orders of the elements ofG, or the set of conjugacy class sizes ofG, or the set of degrees of the irreducible complex characters ofG.

For every setXas above, it is natural to ask to what extent the group structure ofGis reflected and influenced byX, and a useful tool in this kind of investigation is the so-called bipartite divisor graph B(X) of X. The

(*) Indirizzo dell'A.: Department of Mathematics, Gebze Institute of Technol- ogy, Gebze, Turkey.

E-mail: roghayeh@gyte.edu.tr

(**) Indirizzo dell'A.: Dipartimento di Matematica e Applicazioni, University of Milano-Bicocca, Via Cozzi 55 Milano, MI 20125, Italy.

E-mail: pablo.spiga@unimib.it

bipartition of the vertex set ofB(X) consists ofXn f1gand of the set of prime numbers dividingx, for somex2X, and the edge set ofB(X) con- sists of the pairsfp;xgwithgcd(p;x)6ˆ1.

In this paper we are interested in the case thatXconsists of the con- jugacy class sizes ofG, that is,Xˆ fjg^Gj jg2Gg, and hence we study the bipartite divisor graph for the conjugacy class sizes. We will denote this graph simply byB(G). We refer the reader to the beautiful survey [5] for the influence of the conjugacy class sizes on the structure of finite groups.

Historically, various graphs associated with the algebraic structure of a finite group have been extensively studied by a large number of authors, see for example [1, 2, 3, 4, 6, 9, 13, 16]. Recently, Lewis [14] discussed many remarkable connections among these graphs by analysing analogous of these graphs for arbitrary positive integer subsets. Then, inspired by the survey of Lewis, Praeger and Iranmanesh [12] introduced the bipartite divisor graphB(X) for a finite setXof positive integers and studied some basic invariants of this graph (such as the diameter, girth, number of connected components and clique number).

One of the main questions that naturally arises in this area is classifying the groups whose bipartite divisor graphs have special graphical shapes.

For instance, in [10], the first author of this paper and Iranmanesh have classified the groups whose bipartite divisor graphs are paths. Similarly, Taeri [15] considered the case that the bipartite divisor graph is a cycle, and in the course of his investigation posed the following question:

Question([15, Question 1]). Is there any finite group G such that B(G) is isomorphic to a complete bipartite graph K_m;n, for some positive in- tegers m;n2?

The main theorem of this paper gives a family of examples which shows that the answer to this question is positive.

THEOREM1.1. For every odd prime p, there exists af2;pg-group G with B(G)ˆK_2;5.

In light of Theorem 1.1, we pose the following question.

QUESTION1.2. For which positive integers m;n2, is there a finite group G with B(G)ˆKm;n?

In this paper we have tried to produce infinitely many groupsGwith B(G)ˆKm;n,m;n2 and withm;nas small as possible. We were unable

to construct groupsGwithB(G)ˆK2;3. Observe that Dolfi and Jabara [8]

have classified the finite groups having only two non-trivial conjugacy class sizes, and from this remarkable work it follows [8, Corollary C] that there is no groupGwithB(G)ˆK_2;2.

REMARK 1.3. Shortly after the submission of ourarXivpreprint [11]

and during the refereeing process of an earlier draft of this paper, Casolo has provided us of a conjecture addressing Question 1.2. For the sake of completeness we include it here:

CONJECTURE1.4 [7]. If G is a finite group with B(G)ˆK_m;n(where n is the number of non-identity conjugacy class sizes of G), then n2^m.

In particular, Casolo conjectures that there exists no finite group G with B(G)ˆK2;3. (Casolo has proved [7] that there are groups G with B(G)ˆK_2;4, we include a sketch of his construction in Section 3.).

Clearly a ``working'' conjecture can be very useful for the developing of new research, and hence we acknowledge our deepest gratitude to our friend Carlo Casolo for his contribution.

2. Groups with complete bipartite divisor graphs for their conjugacy class sizes

Given a finite group G, we denote byZ(G) the centre ofG, and given g2G, we denote byg^Gthe conjugacy class ofgunderG. Furthermore,jg^Gj denotes the cardinality ofg^G.

NOTATION2.1. We letpbe an odd prime number andEbe the extra- special 2-group of plus-type

Eˆ hx₁;. . .;x_{p 1};y₁;. . .;y_{p 1};z j x²_i ˆy²_i ˆz²ˆ[x_i;z]ˆ[y_i;z]ˆ18i;

[xi;xj]ˆ[yi;yj]ˆ[xi;yj]ˆ18i6ˆj;

[x_i;y_i]ˆz8ii:

Observe that

[x^e₁¹ x^e_p^{p 1}₁;y^h₁¹ y^h_p^p₁¹]ˆz^{e1h1‡ ‡ep 1hp 1}; and that every elementg2Ecan be written uniquely as

gˆx^e₁¹ x^e_p^{p 1}₁y^h₁¹ y^h_p^{p 1}₁zⁿ

withei;h_i2 f0;1gfor eachi2 f1;. . .;p 1gandn2 f0;1g.

Finally, we letAandBbe the maps fromfx1;. . .;xp 1;y1;. . .;yp 1;zg toEgiven (with exponential notation) by

z^Aˆz; z^B ˆz;

x^A_i ˆxi‡1; x^B_i ˆxp i; for eachi2 f1;. . .;p 2g;

x^A_{p 1}ˆx₁ x_{p 1}; x^B_{p 1}ˆx₁;

y^A_i ˆy1yi‡1; y^B_i ˆyp i; for eachi2 f1;. . .;p 2g;

y^A_{p 1}ˆy₁; y^B_{p 1}ˆy₁:

LEMMA2.2. Let E, A and B be as in Notation2:1.Then A and B extend to two automorphisms a and b (respectively) of E.Moreover, a^pˆb²ˆ (ab)²ˆ1andha;biis a dihedral group of order2p.

PROOF. To show thatAandBextend to two automorphisms, sayaandb respectively, ofEit suffices to prove that they preserve the defining relations of E. For instance, for every i2 f1;. . .;p 2g, we have [x^A_i;y^A_i]ˆ [x_i‡1;y₁y_i‡1]ˆ[x_i‡1;y_i‡1]ˆzˆz^Aˆ[x_i;y_i]^A. Similarly, [x^A_{p 1};y^A_{p 1}]ˆ [x1 xp 1;y1]ˆ[x1;y1]ˆzˆz^Aˆ[xp 1;yp 1]^A. Fori;j2 f1;. . .;p 2g with i6ˆj, we have [x^A_i;y^A_j]ˆ[x_i‡1;y1y_j‡1]ˆ1ˆ[x_i;x_j]^A. Moreover, [x^A_i;y^A_{p 1}]ˆ[x_i‡1;y₁]ˆ1ˆ[x_i;y_{p 1}]^Aand [x^A_{p 1};y^A_j]ˆ[x₁ x_{p 1};y₁y_j‡1]ˆ

[x₁;y₁][x_j‡1;y_j‡1]ˆ zzˆ1ˆ[x_{p 1};y_j]^A. All the other computations are similar and are left to the conscientious reader.

It is readily seen thatb²fixes each generatorx1;. . .;xp 1,y1;. . .;yp 1,z of E, and hence b²ˆ1. Observe that x^a_p²₁ˆ(x1 xp 2xp 1)^aˆ x^a₁ x^a_{p 2}x^a_{p 1}ˆx₂ x_{p 1}(x₁ x_{p 1})ˆx₁. Thus x^a₁^p ˆ(x^a₁^{p 2})^a2ˆ x^a_p²₁ˆx₁, and hence x^a_i^p ˆ(x^a₁^{i 1})^ap ˆ(x^a₁^p)^ai1ˆx^a₁ⁱ¹ˆx_i for every i2 f1;. . .;p 1g. Arguing inductively oni, we gety^a_i^{p 1i} ˆy_{p 1 i}y_{p 1}for each i2 f1;. . .;p 1g. It follows that y^a_i^{p i} ˆ(yp i 1yp 1)^aˆy1yp iy1ˆ y_{p i}. Giveni2 f1;. . .;p 1g, applying this equality first to the indexiand then to the indexp i, we get y^a_i^pˆ(y^a_i^{p i})^aiˆy^a_{p i}ⁱ ˆy_{p (p i)} ˆy_i. The- reforea^pfixesx₁;. . .;x_{p 1};y₁;. . .;y_{p 1};zand hencea^pˆ1. Finally, we have

z^abˆz;

x^ab_i ˆx^b_i‡1 ˆx_{p i 1}; 8i2 f1;. . .;p 2g;

x^ab_{p 1}ˆ(x1 xp 1)^bˆxp 1 x1ˆx1. . .xp 1;

y^ab_i ˆ(y1yi‡1)^bˆyp 1yp i 1; 8i2 f1;. . .;p 2g;

y^ab_{p 1}ˆy^b₁ˆy_{p 1};

from which it easily follows that (ab)²ˆ1. p

PROOF OFTHEOREM1:1. Letp;E;AandBbe as in Notation 2.1, leta andbbe the automorphisms extendingAandBas in Lemma 2.2, and set Mˆ ha;bi. Given two elements n1 and n2 of order p, define Nˆ hn₁i hn₂i EandGˆN^jM where

n^a₁ ˆn1; n^a₂ ˆn1n2; n^b₁ˆn1; n^b₂ˆn₂¹

(note that this is well-defined because the action ofMˆ ha;bionhn1;n2i determines a dihedral group of automorphisms of hn1;n2iorder 2p). We show thatB(G)ˆK_2;5.

We start by determining the conjugacy class sizes ofGnZ(G). Clearly, Z(G)ˆ hn₁;zi.

We claim that the conjugacy classes ofGinNnZ(G) have cardinality 2p or 4p. Lete2 fh2Ej[a;h]2Z(E)gand writeeˆx^e₁¹ x^e_p^{p 1}₁y^h₁¹ y^h_p^p₁¹zⁿ, for somee₁;. . .;e_{p 1};h₁;. . .;h_{p 1};n2 f0;1g. We have

e^aˆ(x^e₁¹)^a (x^e_p^{p 1}₁)^a(y^h₁¹)^a (y^h_p^p₁¹)^a(zⁿ)^a

ˆx^e₂¹x^e₃² x^e_p^{p 2}₁(x₁ x_{p 1})^{ep 1}(y₁y₂)^h1(y₁y₃)^h2 (y₁y_{p 1})^{hp 2}y^h₁^{p 1}zⁿ

ˆx^e₁^{p 1}x^e₂^{1‡ep 1} x^e_p^{p 2}₁^{‡ep 1}y^h₁^{1‡ ‡hp 1}y^h₂¹ y^h_p^p₁²zⁿ:

As e ¹e^aˆ[e;a]2Z(E)ˆ hzi, by comparing e with e^a, we get e_iˆ h_iˆ0 for every i2 f1;. . .;p 1g, and hence e2Z(E). Thus fh2Ej[a;h]2Z(E)g ˆZ(E). In particular,C_E…a† ˆZ(E) andaacts fixed point freely onEnZ(E). It follows that the orbits ofMacting by conjuga- tion onEnZ(E) have size por 2p. SincejE⁰j ˆ2, for everye2EnZ(E) we havejE:C_E…e†j ˆ2. Thuse^Ghas size 2por 4p. Now letg2NnZ(N).

Writegˆne, for some n2 hn₁;n₂i n hn₁iand e2En hzi. Now C_N…g† ˆ hn₁;n₂iC_E…e†has index 2 in Nand so (asadoes not centralise n)C_G…g†

has index divisible by 2p. It follows (with an elementary computation) that the conjugacy classes of Gin NnZ(G) have size 2p or 4p. Observe that both possibilities can occur: jx^G₁j ˆ4p andj(x1xp 1)^Gj ˆ2p.

Let g2 hN;ai nN and write gˆnaⁱ, for some n2N and i2

f1;. . .;p 1g. Now, givene2E, we haveeⁿˆezⁿ, for somen2 f0;1g. In

particular, if e2CE…g†, then eˆe^gˆ(eⁿ)^aiˆ(ezⁿ)^aiˆe^aizⁿ, and hence [a;e]2Z(E). Thus e2Z(E). It follows thatCN…g† ˆZ(G) and jC_G…g†j ˆ pjCN…g†j ˆ2p². Thusjg^Gj ˆ2^{2p 1}p.

It remains to compute the size of the conjugacy classes of G in Gn ha;Ni. Observe that every element of Gn ha;Ni is conjugate to an element of hb;Ni nN. In particular, since we are interested only on the conjugacy class sizes, we may assume thatg2 hb;Ni nN. Writegˆnb, for

somen2N. Observe that

fh2Ej[h;b]2Z(E)g ˆhx1xp 1;x2xp 2;. . .;x_{(p 1)=2}x_(p‡1)=2;

y₁y_{p 1};y₂y_{p 2};. . .;y_{(p 1)=2}y_(p‡1)=2;zi ˆC_E…b†;

and jC_E…b†j ˆ2^p. Now, arguing as above, it is easy to verify that C_N…g† hn₁;C_E…b†i and jhn₁;C_E…b†i:C_N…g†j 2 f1;2g. Thus jC_G…g†j 2 fp²2^p;p²2^p‡1g and hencejg^Gjequals 2^{p 1}p² or 2^pp². Observe that both of these cases can occur: in fact jb^Gj ˆ jG:C_G…b†j ˆ2^{p 1}p² and j(x1b)^Gj ˆ jG:C_G…x1b†j ˆ2^pp².

Summing up, the conjugacy class sizesjg^Gj(asgruns throughGnZ(G)) are 2p;4p;2^{2p 1}p;2^{p 1}p²;2^pp², from which it follows thatB(G)ˆK_2;5. p

3. The caseB(G)ˆK_2;4

Here we describe a groupGwithB(G)ˆK2;4: the construction is due to Casolo [7]. LetQ be the quaternion group of order 8, and letQ₁ˆ hq₁i, Q₂ˆ hq₂iandQ₃ˆ hq₃ibe the three maximal subgroups ofQ. Letpbe an odd prime and letEbe the extra-specialp-group

hx₁;x₂;x₃;y₁;y₂;y₃;zjx^p_i ˆy^p_i ˆz^pˆ[x_i;z]ˆ[y_i;z]ˆ18i;

[x_i;x_j]ˆ[y_i;y_j]ˆ[x_i;y_j]ˆ18i6ˆj;[x_i;y_i]ˆz8ii:

Arguing as in Section 2, it is easy to verify that by assigning

x^q₁¹ˆx1; y^q₁¹ˆy1; x^q₂¹ˆx₂¹; y^q₂¹ˆy₂¹; x^q₃¹ˆx₃¹; y^q₃¹ˆy₃¹; z^q1ˆz;

x^q₁²ˆx₁¹; y^q₁²ˆy₁¹; x^q₂²ˆx2; y^q₂²ˆy2; x^q₃²ˆx₃¹; y^q₃²ˆy₃¹; z^q2ˆz;

x^q₁³ˆx₁¹; y^q₁³ˆy₁¹; x^q₂³ˆx₂¹; y^q₂³ˆy₂¹; x^q₃³ˆx3; y^q₃³ˆy3; z^q3ˆz;

we define a group-action ofQonE. SetGˆE^jQ.

Now, following the proof of Theorem 1.1 we see that the conjugacy class sizesjg^Gj, asgruns throughGnZ(G), are 2p, 4p, 2p⁴and 2p⁵. In particular, B(G)ˆK_2;4.

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Manoscritto pervenuto in redazione il 22 settembre 2013.