The classification and the conjugacy classes of the finite subgroups of the sphere braid groups

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The classification and the conjugacy classes of the finite subgroups of the sphere braid groups

Daciberg Gonçalves, John Guaschi

To cite this version:

Daciberg Gonçalves, John Guaschi. The classification and the conjugacy classes of the finite subgroups of the sphere braid groups. Algebraic and Geometric Topology, Mathematical Sciences Publishers, 2008, 8 (2), pp.757-785. �10.2140/agt.2008.8.757�. �hal-00191422�

hal-00191422, version 1 - 26 Nov 2007

The classification and the conjugacy classes of the finite subgroups of the sphere braid groups

DACIBERG LIMA GONC ¸ ALVES Departamento de Matem´atica - IME-USP, Caixa Postal 66281 - Ag. Cidade de S˜ao Paulo,

CEP: 05311-970 - S˜ao Paulo - SP - Brazil.

e-mail: dlgoncal@ime.usp.br JOHN GUASCHI

Laboratoire de Math´ematiques Nicolas Oresme UMR CNRS 6139, Universit´e de Caen BP 5186,

14032 Caen Cedex, France.

e-mail: guaschi@math.unicaen.fr 21st November 2007

Abstract

Let n^¥3. We classify the finite groups which are realised as subgroups of the sphere braid group Bn^pS^2q. Such groups must be of cohomological period 2 or 4. Depending on the value of n, we show that the following are the maximal finite subgroups of Bn^pS^2q:Z₂pn1^q; the dicyclic groups of order4n and4^pn2^q; the binary tetrahedral group T1; the binary octahedral group O1; and the binary icosahedral group I. We give geometric as well as some explicit algebraic constructions of these groups in Bn^pS^2q, and determine the number of conjugacy classes of such finite subgroups. We also reprove Murasugi’s classification of the torsion elements of Bn^pS^2q, and explain how the finite subgroups of Bn^pS^2qare related to this classification, as well as to the lower central and derived series of Bn^pS^2q.

1 Introduction

The braid groups B_n of the plane were introduced by E. Artin in 1925 [A1, A2]. Braid groups of surfaces were studied by Zariski [Z]. They were later generalised by Fox to braid groups of arbitrary topological spaces via the following definition [FoN]. Let M be a compact, connected surface, and letn ^PN. We denote the set of all orderedn-tuples of distinct points ofM, known as then^th configuration space of M, by:

F_n^pM^q

!

pp₁, . . . ,p_n^q

p_i ^P Mand p_i p_j ifi j

)

.

2000 AMS Subject Classification: 20F36 (primary), 20F50, 20E45, 57M99 (secondary).

Configuration spaces play an important r ˆole in several branches of mathematics and have been extensively studied, see [CG, FH] for example.

The symmetric groupS_nonnletters acts freely onF_n^pM^qby permuting coordinates.

The corresponding quotient will be denoted byD_n^pM^q. Then^thpure braid group P_n^pM^q (respectively then^thbraid group B_n^pM^q) is defined to be the fundamental group ofF_n^pM^q (respectively of D_n^pM^q).

Together with the real projective plane RP², the braid groups of the 2-sphere S² are of particular interest, notably because they have non-trivial centre [GVB, GG1], and torsion elements [VB, Mu]. Indeed, Van Buskirk showed that among the braid groups of compact, connected surfaces, B_n^pS^2qand B_n^pRP^2qare the only ones to have torsion [VB]. Let us recall briefly some of the properties of B_n^pS^2q[FVB, GVB, VB].

If D^{2 „} S² is a topological disc, there is a group homomorphism ι: B_n ^ÝÑB_n^pS^2q induced by the inclusion. If β ^P B_n, we shall denote its imageι^pβ^q simply byβ. Then B_n^pS^2qis generated byσ₁, . . . ,σ_n1which are subject to the following relations:

σ_iσ_j σ_jσ_i if|ij|^¥2 and 1^¤i,j ^¤n1 σ_iσ_{i 1}σ_i σ_{i 1}σ_iσ_{i 1}for all 1^¤i^¤n2, and

σ₁σ_n2σ_n²1σ_n2σ₁1.

Consequently, B_n^pS^2qis a quotient ofB_n. The first three sphere braid groups are finite:

B₁^pS^2qis trivial,B₂^pS^2qis cyclic of order 2, andB₃^pS^2qis a ZS-metacyclic group (a group whose Sylow subgroups, commutator subgroup and commutator quotient group are all cyclic) of order 12, isomorphic to the semi-direct productZ₃Z₄of cyclic groups, the action being the non-trivial one, which in turn is isomorphic to the dicyclic group Dic₁₂ of order 12. The Abelianisation ofB_n^pS^2qis isomorphic to the cyclic groupZ₂^p_n1^q. The kernel of the associated projectionξ: B_n^pS^2qÝÑZ₂pn1^q (which is defined byξ^pσ_i^q1 for all 1 ^¤i ^¤ n1) is the commutator subgroupΓ₂

B_n^pS^2q . If w ^P B_n^pS^2qthenξ^pw^q is the exponent sum (relative to theσ_i) ofwmodulo 2^pn1^q.

Gillette and Van Buskirk showed that ifn^¥3 andk^P NthenB_n^pS^2qhas an element of order k if and only if k divides one of 2n, 2^pn1^q or 2^pn2^q [GVB]. The torsion elements ofB_n^pS^2qandB_n^pRP^2qwere later characterised by Murasugi [Mu]. ForB_n^pS^2q, these elements are as follows:

THEOREM 1 ([MU]). Let n ^¥ 3. Then the torsion elements of B_n^pS^2qare precisely powers of conjugates of the following three elements:

(a) α0σ₁σn2σ_n1(which is of order2n).

(b) α₁ σ₁σ_n2σ_n²

1(of order2^pn1^q).

(c) α2 σ₁σn3σ_n²2(of order2^pn2^q).

The three elementsα₀, α₁andα₂are respectivelyn^th,^pn1^qthand^pn2^qthroots of

∆_n, where∆_n is the so-called ‘full twist’ braid ofB_n^pS^2q, defined by∆_n pσ₁σ_n1^qn. SoB_n^pS^2qadmits finite cyclic subgroups isomorphic toZ_2n,Z₂^p_n1^qandZ₂^p_n2^q. In [GG2], we showed that B_n^pS^2q is generated byα₀ and α₁. Ifn ^¥ 3, ∆_n is the unique element of B_n^pS^2q of order 2, and it generates the centre of B_n^pS^2q. It is also the square of the Garside element(or ‘half twist’) defined by:

T_n ^pσ₁σ_n1^qpσ₁σ_n2^qpσ₁σ₂^qσ₁.

Forn^¥4,B_n^pS^2qis infinite. It is an interesting question as to which finite groups are realised as subgroups of B_n^pS^2q(apart of course from the cyclic groups ^xα_i^y and their subgroups given in Theorem 1). Another question is the following: how many conju- gacy classes are there inB_n^pS^2qof a given abstract finite group? As a partial answer to the first question, we proved in [GG2] that B_n^pS^2qcontains an isomorphic copy of the finite group B₃^pS^2qof order 12 if and only ifn 1 mod 3.

While studying the lower central and derived series of the sphere braid groups, we showed that Γ₂

B₄^pS^2q is isomorphic to a semi-direct product of Q₈ by a free group of rank 2 [GG3]. After having proved this result, we noticed that the question of the realisation ofQ₈as a subgroup of B_n^pS^2qhad been explicitly posed by R. Brown [ATD]

in connection with the Dirac string trick [F, N] and the fact that the fundamental group of SO^p3^qis isomorphic toZ₂. The casen4 was studied by J. G. Thompson [ThJ]. In a previous paper, we provided a complete answer to this question:

THEOREM 2 ([GG4]).Let n^PN, n^¥3.

(a) B_n^pS^2qcontains a subgroup isomorphic toQ₈if and only if n is even.

(b) If n is divisible by4thenΓ₂

B_n^pS^2q contains a subgroup isomorphic toQ₈.

As we also pointed out in [GG4], for alln^¥3, the construction ofQ₈may be gener- alised in order to obtain a subgroup^xα₀,T_n^yofB_n^pS^2qisomorphic to the dicyclic group Dic_4n of order 4n.

It is thus natural to ask which other finite groups are realised as subgroups ofB_n^pS^2q. One common property of the above subgroups is that they are finite periodic groups of cohomological period 2 or 4. In fact, this is true for all finite subgroups of B_n^pS^2q. Indeed, by [GG2], the universal covering X of F_n^pS^2q is a finite-dimensional complex which has the homotopy type of S³ (we were recently informed by V. Lin that X is biholomorphic to the direct product of SL^p2,C^q by the Teichm ¨uller space of the n- punctured Riemann sphere [Li]). Thus any finite subgroup of B_n^pS^2q acts freely on X, and so has period 2 or 4 by Proposition 10.2, Section 10, Chapter VII of [Br]. Since

∆_n is the unique element of order 2 of B_n^pS^2q, and it generates the centre Z^pB_n^pS^2qq, the Milnor property must be satisfied for any finite subgroup of B_n^pS^2q. Recall also that a finite periodic group G satisfies the p²-condition (if p is prime and divides the order ofGthenGhas no subgroup isomorphic toZ_pZ_p), which implies that a Sylow p-subgroup of G is cyclic or generalised quaternion, as well as the 2p-condition (each subgroup of order 2p is cyclic). The classification of finite periodic groups is given by the Suzuki-Zassenhaus theorem (see [AM, ThC] for example), and thus provides a possible line of attack for the subgroup realisation problem. The periods of the differ- ent families of these groups were determined in a series of papers by Golasi ´nski and Gonc¸alves [GoG1, GoG2, GoG3, GoG4, GoG5, GoG6], and so in theory we may obtain a list of those of period 4. A list of all periodic groups of period 4 is provided in [ThC].

However, in the current context, a more direct approach is obtained via the relation- ship between the braid groups and the mapping class groups of S², which we shall now recall.

Forn ^P N, letM_0,n denote the mapping class group of then-punctured sphere. We allow thenmarked points to be permuted. Ifn^¥2, a presentation ofM_0,n is obtained from that of B_n^pS^2qby adding the relation∆_n 1 [Ma, MKS]. In other words, we have

the following central extension:

1^ÝÑx∆_nyÝÑB_n^pS^2qÝpÑM_0,n ^ÝÑ1. (1) If n 2, B₂^pS^2q M_0,2 Z₂. For n 3, since M_0,3 S₃, this short exact sequence does not split, and in fact forn ^¥4 it does not split either [GVB].

This exact sequence may also be obtained in the following manner [Bi]. Let Diff

S² denote the group of orientation-preserving homeomorphisms ofS², and letX^P D_n^pS^2q. Then Diff

S²,X

"

f ^P Diff

S² f^pX^qX

*

is a subgroup of Diff

S² , and we have a fibration Diff

S²,X ^ÝÑDiff

S^{2 ÝÑ} D_n^pS^2q, where the basepoint of D_n^pS^2q is taken to be X, and where the second map evaluates an element of Diff

S² on X.

The resulting long exact sequence in homotopy yields:

ÝÑπ₁

Diff

S²,X

ÝÑπ₁

Diff

S²

loooooooooomoooooooooon

Z₂

ÝÑπlooooooomooooooon₁^pD_n^pS^2qq

Bn^pS^2q δ

ÝÑπ₀

Diff

S²,X

looooooooooooomooooooooooooon

M_0,n

ÝÑπ₀

Diff

S²

loooooooooomoooooooooon

t1^u

. (2)

The homomorphism δ: B_n^pS^2qÝÑM_0,n is the boundary operator which we shall use in Section 3 in order to describe the geometric realisation of the finite subgroups of B_n^pS^2q. If n ^¥ 3 thenπ₁

Diff

S²,X

t1^u[EE, Hm, Sc], and we thus recover equa- tion (1) (the interpretation of the Dirac string trick in terms of the sphere braid groups [F, Hn, N] gives rise to the identification ofπ₁

Diff

S²

with ^x∆_ny).

In a recent paper, Stukow applies Kerckhoff’s solution of the Nielsen realisation problem [K] to classify the finite maximal subgroups ofM_0,n [St]. Applying his results to equation (1), we shall see in Section 2 that their counterparts in B_n^pS^2q are cyclic, dicyclic and binary polyhedral groups:

THEOREM 3.Let n^¥3. The maximal finite subgroups of B_n^pS^2qare:

(a) Z₂pn1^qif n^¥5.

(b) the dicyclic group Dic_4n of order4n.

(c) the dicyclic group Dic₄pn2^q if n5or n^¥7.

(d) the binary tetrahedral group, denoted by T₁, if n4 mod 6.

(e) the binary octahedral group, denoted by O₁, if n0, 2 mod 6.

(f) the binary icosahedral group, denoted by I, if n 0, 2, 12, 20 mod 30.

REMARKS 4.

(a) Ifnis odd then the only finite subgroups ofB_n^pS^2qare cyclic or dicyclic. In the latter case, the dicyclic group Dic_4n(resp. Dic₄pn2^q) is ZS-metacyclic [CM], and is isomorphic toZnZ₄(resp.Z_n2Z₄), where the action is multiplication by1.

(b) Ifn is even then one of the binary tetrahedral or octahedral groups is realised as a maximal finite subgroup ofB_n^pS^2q. Further, since T₁is a subgroup ofO₁, T₁is realised as a subgroup ofB_n^pS^2qfor allneven,n^¥4.

(c) The groups of Theorem 3 and their subgroups are the finite groups of quaternions [Co].

Indeed, for p,q,r^P N, let us denote

xp,q,r^{y x}A,B,C |A^p B^q C^r ABC^y.

ThenZ₂pn1^{q x}n1,n1, 1^y, Dic_4n ^xn, 2, 2^y, Dic₄pn2^{q x}n2, 2, 2^y, T₁ ^x3, 3, 2^y, O₁ ^x4, 3, 2^y and I ^x5, 3, 2^y. It is shown in [Co, CM] that for T₁, O₁ and I, this presentation is equivalent to:

xp, 3, 2^y

B

A,B

A^p B^3pAB^q2

F

,

for p^{P t}3, 4, 5^u, and that the element A^pis central and is the unique element of order 2 of^xp, 3, 2^y.

In Section 2, we also generalise another result of Stukow concerning the conjugacy classes of finite subgroups ofM_0,n toB_n^pS^2q:

PROPOSITION 5.

(a) Two maximal finite subgroups of B_n^pS^2qare isomorphic if and only if they are conjugate.

(b) Each abstract finite subgroup G of B_n^pS^2q is realised as a single conjugacy class within B_n^pS^2q, with the exception, when n is even, of the following cases, for which there are precisely two conjugacy classes:

(i) G Z₄.

(ii) G Dic_4r, where r divides ⁿ₂ or ⁿ₂².

In Section 3, we explain how to obtain geometrically the subgroups of Theorem 3, and we also give explicit group presentations of the cyclic and dicyclic subgroups, as well as in the special caseT₁forn4.

In order to understand better the finite subgroups of B_n^pS^2q, it is often useful to know their relationship with the three classes of elements described in Theorem 1. This shall be carried out in Proposition 15 (see Section 4).

The two conjugacy classes of part (b)(i) are realised by the subgroups

B

αⁿ₀^{2

F

and

B

α^p₂^n2q{2

F

(they are non conjugate since they project to non-conjugate subgroups in S_n). In Section 5, we construct the two conjugacy classes of part (b)(ii) of Proposition 5:

THEOREM 6.Let n ^¥ 4 be even. Let N ^{P t}n,n2^u, and let x α₀ (resp. x α₀α₂α₀¹) if N n (resp. N n2). Set N 2^lk, where l ^PN, and k is odd. Then for j0, 1, . . . ,l, and q a divisor of k, we have:

(a) B_n^pS^2qcontains2^jcopies of Dic₂l 2jk^{qof the form

A

x^2jq,x^iqT_n

E

, where i0, 1, . . . , 2^j1.

(b) if 0 ^¤ i,i^{1 ¤} 2^j1,

A

x^2jq,x^iqT_n

E

and

A

x^2jq,x^i1qT_n

E

are conjugate if and only if ii¹ is even.

Another question arising from Theorem 2 is the existence of copies of Q₈ lying in Γ₂pB_n^pS^2qq. More generally, one may ask whether the dicyclic groups constructed above (and indeed the other finite subgroups of B_n^pS^2q) are contained in Γ₂pB_n^pS^2qq. In the dicyclic case, we have the following result, also proved in Section 5:

PROPOSITION 7.Let n ^¥ 4 be even, let N ^{P t}n,n2^u, and let r divide N. If r does not divide N^{2 then the subgroups of B_n^pS^2q abstractly isomorphic to Dic_4r are not contained in Γ₂pB_n^pS^2qq. If r divides N^{2 then up to conjugacy, B_n^pS^2q has a two subgroups abstractly isomorphic to Dic_4r, one of which is contained inΓ₂pB_n^pS^2qq, and the other not. In particular, B_n^pS^2qexhibits the two conjugacy classes ofQ₈, one of which lies inΓ₂pB_n^pS^2qq, the other not.

The corresponding result for the binary polyhedral groups may be found in Propo- sition 16. As a corollary of our results we obtain an alternative proof of Theorem 1 (see Section 6).

Acknowledgements

This work took place during the visits of the second author to the Departmento de Matem´atica do IME-Universidade de S˜ao Paulo during the periods 10^th – 27^thAugust 2007 and 4^th – 23^rd October 2007, and of the visit of the first author to the Laboratoire de Math´ematiques Emile Picard, Universit´e Paul Sabatier, Toulouse, during the period 27^th August – 17^th September 2007. It was supported by the international Cooperation USP/Cofecub project number 105/06, by an ‘Aide Ponctuelle de Coop´eration’ from the Universit´e Paul Sabatier, by the Pr ´o-Reitoria de Pesquisa Projeto 1, and by the Projeto Tem´atico FAPESP no. 2004/10229-6 ‘Topologia Alg´ebrica, Geometrica e Diferencial’.

2 The classification of the finite maximal subgroups of B

S

In this section, we prove Theorem 3. We start by making some remarks concerning the central extension (1).

REMARKS 8. LetGbe a finite subgroup ofB_n^pS^2q.

(a) IfHis a finite subgroup ofM_0,n then p^1pH^qis a finite subgroup ofB_n^pS^2qof order 2|H|.

(b) If |G| is odd then ∆_n R G, and so G p^pG^q. Conversely, if G p^pG^q then p^|_G is injective, and thus∆_n R G, so|G|is odd.

(c) If|G|is even then∆_n P G, and so we obtain the following short exact sequence:

1^ÝÑx∆_nyÝÑG ^ÝpÑ|G p^pG^qÝÑ1, (3) where p^pG^qis a finite subgroup ofM_0,n of order |G|

2 .

(d) IfGis a maximal finite subgroup of B_n^pS^2qthen|G|is even, and p^pG^qis a maximal finite subgroup of M_0,n. Conversely, if H is a maximal finite subgroup of M_0,n then p^1pH^qis a maximal finite subgroup ofB_n^pS^2q.

We recall Stukow’s theorem:

THEOREM 9 ([ST]).Let n^¥3. The maximal finite subgroups ofM_0,n are:

(a) Z_n1if n4.

(b) the dihedral group D_2n of order2n.

(c) the dihedral group D₂pn2^q if n5or n^¥7.

(d) A₄if n4, 10 mod 12.

(e) S₄if n0, 2, 6, 8, 12, 14, 18, 20 mod 24.

(f) A₅if n0, 2, 12, 20, 30, 32, 42, 50 mod 60.

REMARK 10. In the case n 3, M_0,3 is isomorphic to D₆, obtained as a maximal sub- group in part (b) of Theorem 9, and so its subgroup isomorphic toZ₂ is not maximal.

This explains the discrepancy between the value ofnin part (a) of Theorems 3 and 9.

Proof of Theorem 3. By Remarks 8, we just need to check that the given groups are those obtained as extensions of^x∆_nyby the groups of Theorem 9. We start by making some preliminary remarks. Let H be one of the finite maximal subgroups of M_0,n, and let G be a finite (maximal) subgroup of B_n^pS^2qof order 2|H|which fits into the following short exact sequence:

1^ÝÑx∆_ny ÝÑG^ÝpÑ|G H^ÝÑ1, (4) where ∆_n P G belongs to the centre of G, and is the unique element of G of order 2.

ThenG p^1pH^q, and so is unique.

Suppose thaty ^P H is of orderk ^¥2. Thenyhas two preimages inG, of the formx and x∆_n, say, and xis of order kor 2k. If k is even then by Remarks 8(c),x must be of order 2k, x^k ∆_n and ∆_n P xx^y. If kis odd then x is of orderk (resp. 2k) if and only if x∆_n is of order 2k(resp.k).

A presentation of G may be obtained by applying standard results concerning the presentation of an extension (see Theorem 1, Chapter 13 of [J]). If H is generated by h₁, . . . ,h_k then G is generated by g₁, . . . ,g_k,∆_n, where p^pg_i^q h_i fori 1, . . . ,k. One relation ofGis just∆²

n 1, that of Ker^pp^q. Since Ker^pp^q„Z^pG^q, the remaining relations of G are obtained by rewriting the relators of H in terms of the coset representatives, and expressing the corresponding element in the form∆^ε

n, whereε^Pt0, 1^u. We consider the six cases of Theorem 9 as follows.

(a) H Z_n1: let y be a generator of H, and let x ^P G be such that p^px^q y. Then G ^x∆_n,x^y and |G| 2^pn1^q. If nis odd then ∆_n P xx^y, G ^xx^y, and x is of order 2^pn1^q. Ifnis even thenG^xx∆_ny(resp.G^xx^y) ifxis of ordern1 (resp. 2^pn1^q), andG Z₂pn1^q in both cases.

(b) H D_2n: let y,z ^P H be such that o^py^q n, o^pz^q 2 and zyz¹ y¹, and let x,w ^P G be such that p^px^q y and p^pw^q z. So G ^x∆_n,x,w^yand |G| 4n. From above, it follows that w² ∆_n, so G ^xx,w^y. If n is even then x is of order 2n and xⁿ ∆_n. The same result may be obtained ifnis odd, replacingxby x∆_n if necessary.

Further, wxw¹x ^P Ker^pp^q. If wxw¹x ∆_n then ^pwx^q2 1. So either w x¹ or wx ∆_n, and in both cases we conclude that G ^xx^y which contradicts |G| 4n.

Hencewxw¹x1, and since |G|4n,Gis isomorphic to Dic_4n. (c) H D₂pn2^q: the previous argument shows thatGDic₄pn2^q.

(d) Suppose that His isomorphic to one of the remaining groups A₄, S₄ or A₅ of The- orem 9. Let p 3 if H A₄, p 4 if H S₄, and p 5 if H A₅. Then H has a presentation given by [Co, CM]:

H

B

u,v

u²v^{3 p}uv^qp1

F

.

Let x,w ^P G be such that p^px^q u and p^pw^q v. Then G ^xx,w,∆_ny. From above, we must have x² ∆_n. Further, replacingw byw∆_n, we may suppose thatw³ ∆_n.

If p 4 then^pxw^qp ∆_n, while if p ^{P t}3, 5^u, replacing x by x∆_n if necessary, we may suppose that^pxw^qp ∆_n. It is shown in [Co, CM] that x² w^{3 p}xw^qp ∆_n implies that∆²

n 1, soGadmits a presentation given by:

G

B

x,w

x²w^{3 p}xw^qp

F

.

ThusG T₁if p 3, G O₁if p 4 andG I if p 5. This completes the proof of the theorem.

REMARKS 11. LetG₁,G₂be finite subgroups ofB_n^pS^2q.

(a) If they are of odd order then by Remarks 8,G₁andG₂are isomorphic if and only if p^pG₁^qand p^pG₂^qare isomorphic. So suppose thatG₁ andG₂are of even order. If p^pG₁^q and p^pG₂^q are isomorphic then it follows from the construction of Theorem 3 thatG₁ andG₂are isomorphic. Conversely, suppose thatG₁and G₂are isomorphic via an iso- morphismα: G₁ ^ÝÑG₂. Since∆_nbelongs to both, and is the unique element of order 2, we must haveα^p∆_nq∆_n, and thusαinduces an isomorphismα^r: p^pG₁^qÝÑp^pG₂^qsat- isfying^rαp pα.

(b) If G₁,G₂ are conjugate then clearly so are p^pG₁^q and p^pG₂^q. Conversely, suppose that p^pG₁^q,p^pG₂^q are conjugate subgroups of M_0,n. Then there exists g ^P M_0,n such that p^pG₂^q gp^pG₁^qg¹. IfG₁ and G₂ are of even order, the fact that equation (1) is a central extension implies that G₁,G₂ are conjugate. IfG₁ and G₂ are of odd order, let L_i p^1pp^pG_i^qqfori 1, 2. Then^rL_i : G_i^s 2, and it follows from the even order case that L₁and L₂are conjugate in B_n^pS^2q. ButL_i G_i^²∆_nG_i, and its odd order elements are precisely those ofG_i. So the conjugacy betweenL₁andL₂ must sendG₁ontoG₂.

We are now able to prove Proposition 5.

Proof of Proposition 5. Part (a) follows from Remarks 8 and 11. To prove part (b), let G₁,G₂ be abstractly isomorphic finite subgroups of Bn^pS^2q, and for i 1, 2, let H_i p^pG_i^q. ThenH₁ H₂: if theG_i are of odd order thenH_i G_i, so H₁ H₂, while if the G_iare of even order, any isomorphism between them must send ∆_n P G₁onto∆_n P G₂, and so projects to an isomorphism between the H_i. From Remarks 11(b), G₁ and G₂ are conjugate if and only if H₁ and H₂ are, and so the number of conjugacy classes of subgroups ofB_n^pS^2qisomorphic toG₁is the same as the number of conjugacy classes of subgroups of M_0,n isomorphic to H₁. The result follows from the proof of Theorem 3 by remarking that a subgroup ofM_0,n isomorphic toZ₂(resp. D_2r) lifts to a subgroup ofB_n^pS^2qwhich is isomorphic toZ₄(resp. Dic_4r).

3 Realisation of the maximal finite subgroups of B

S

In this section, we analyse the geometric and algebraic realisations of the subgroups given in Theorem 3.

3.1 The algebraic realisation of some finite subgroups of B

S

The maximal cyclic and dicyclic subgroups ofB_n^pS^2qmay be realised as follows: