ON TWISTED HOMOGENEOUS RACKS OF TYPE D. Dedicado a Hans-Jürgen Schneider en ocasión de sus 65 años.

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1 REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Voumen 51, Número 2, 2010, Páginas 1 16 ON TWISTED HOMOGENEOUS RACKS OF TYPE D N. ANDRUSKIEWITSCH, F. FANTINO, G. A. GARCÍA AND L. VENDRAMIN Abstract. We deveop some techniques to check when a twisted homogeneous rack of cass (L,t,θ) is of type D. Then we appy the obtained resuts to the cases L = A n, n 5, or L a sporadic group. Dedicado a Hans-Jürgen Schneider en ocasión de sus 65 años. 1. Introduction A rack is a pair (X, ) where X is a non-empty set and : X X X is an operation such that the map ϕ x = x is bijective for any x X, and x (y z) = (x y) (x z) for a x,y,z X. For instance, a conjugacy cass in a group is a rack with given by conjugation. We are interested in the foowing cass of racks. Definition 1.1. A rack X is of type D if there exists a decomposabe subrack Y = R S of X such that r (s (r s)) s, for some r R,s S. (1) Racks appear in the cassification probem of finite-dimensiona pointed Hopf agebras over C. Let us expain very briefy this reation. A finite-dimensiona pointed Hopf agebra H has attached some basic invariants: a rack X, n N and a 2-cocyce q : X X GL(n, C); these satisfy that the Nichos agebra B(X, q) has finite dimension. This has been expained at ength in severa papers, see [AG, AFGV1, AFGV2]. We refer to [AS] for a detaied exposition of the notion of Nichos agebra of a Yetter-Drinfed modue, of crucia importance in this project. Therefore, a fundamenta step for the cassification probem is to sove the foowing question. Question 1. For any finite rack X, and for any cocyce q, determine if B(X, q) is finite-dimensiona. This question is reated to racks of type D by the foowing resut, a reinterpretation of [HS, Thm. 8.6], in turn a consequence of [AHS]. Theorem 1.2. [AFGV1, Th. 3.6] If X is a finite rack of type D, then X coapses, that is dim B(X, q) = for any q Mathematics Subject Cassification. 16T05; 17B37. This work was partiay supported by ANPCyT-Foncyt, CONICET, Ministerio de Ciencia y Tecnoogía (Córdoba), Secyt-UNC and Secyt-UBA. 1

2 2 ANDRUSKIEWITSCH, FANTINO, GARCÍA, AND VENDRAMIN Consequenty, it is natura to address the foowing question. Question 2. Determine a simpe racks of type D. Indeed, any indecomposabe rack Z admits a rack epimorphism π : Z X with X simpe; and if X is of type D, then Z is of type D. Now, finite simpe racks have been cassified in [AG, 3.9, 3.12], see aso [J]. Succincty, any simpe rack beongs to one of the foowing casses: (1) Simpe affine racks. (2) Non-trivia (twisted) conjugacy casses in simpe groups. (3) Simpe twisted homogeneous racks of cass (L, t, θ); these are twisted conjugacy casses corresponding to (G,u), where G = L t, with L a finite simpe non-abeian group, t > 1 and θ Aut(L), u Aut(L t ) acts by See Subsection 3.2 beow. u( 1,..., t ) = (θ( t ), 1,..., t 1 ), 1,..., t L. The goa of the present paper is to study when a simpe twisted homogeneous rack, that is case (3), is of type D. We first show that twisted homogeneous racks (THR, for short) of cass (L, t, θ) are parameterized by twisted conjugacy casses in L with respect to θ, see Proposition 3.3. For exampe, assume that L = A n, n 5, and θ = ι (1 2) (conjugation in S n ). Then the THR of type (A n,t,ι (1 2) ) are parameterized by the conjugacy casses in S n not contained in A n ; this expains the notation in Tabe 2. Then we deveop some techniques to check that a twisted homogeneous rack is of type D, see Section 3; neither of these resuts requires the simpicity of L. Our main resuts are proved in Section 4: Theorem 1.3. Let L be A n, n 5, θ Aut(L), t 2 and L. If C is a twisted homogeneous rack of cass (L,t,θ) not isted in Tabes 1, 2, then C is of type D. Theorem 1.4. Let L be a sporadic group, θ = id, t 2 and L. If C is a twisted homogeneous rack of cass (L,t,θ) not isted in Tabe 3, then C is of type D. We have a computer program based on GAP to check that a conjugacy of some group is of type D. Some of us are presenty trying to dea with the inconcusive resuts as to the racks in the theorems above are of type D or not. Athough there are infinitey many such inconcusive cases of THR unknown to be of type D, and therefore the probem coud not be soved merey by using this computer program, we do expect that the treatment of some exampes woud iuminate and give us a hint to more genera cases. Aas, the program takes a ot of time and even some simper cases ike L = A 5, = e and t = 4 are very hard and we sti do not have an answer. The case: L a sporadic group, θ id, wi be treated in another paper. See however a preiminary discussion in Subsection 4.4. Rev. Un. Mat. Argentina, Vo 51-2

3 TWISTED HOMOGENEOUS RACKS OF TYPE D 3 Tabe 1. THR C of type (A n,t,θ), θ = id, t 2, n 5, not known of type D. Those not of type D are in bod. n Type of t any e (1 n ) odd, (t,n!) = 1 5 (1 5 ) 2 5 (1 5 ) 4 6 (1 6 ) 2 5 invoution (1,2 2 ) 4, odd 6 (1 2,2 2 ) odd 8 (2 4 ) odd any order 4 (1 r1,2 r2,4 r4 ), r 4 > 0, r 2 +r 4 even 2 Tabe 2. THR C of type (A n,t,θ), θ = ι (12), t 2, n 5, not known of type D. n Type of (1 2) t any (1 s1,2 s2,...,n sn ), s 1 1 and s 2 = 0 any s h 1, for some h, 3 h n (1 s1,2 s2,4 s4 ), s 1 2 or s 2 1, 2 s 2 +s 4 odd, s (1 3,2) 2, 4 6 (1 4,2) 2 (2 3 ) 2 7 (1,2 3 ) 2, odd 8 (1 2,2 3 ) odd 10 (2 5 ) odd Tabe 3. THR C of type (L,t,θ), with L a sporadic group, θ = id, not known of type D. sporadic group Type of or t cass name of O L any 1A (t, L ) = 1, t odd ord() = 4 2 T, J 2, Fi 22, Fi 23, Co 2 2A odd B 2A, 2C odd Suz 6B, 6C any Rev. Un. Mat. Argentina, Vo 51-2

4 4 ANDRUSKIEWITSCH, FANTINO, GARCÍA, AND VENDRAMIN 2. Preiminaries 2.1. Gossary. If t,n N, then (t,n) denotes their highest common divisor. If G is a group and x G, then x denotes the cycic subgroup generated by x; ι x, the inner automorphism associated to x; O G x, the conjugacy cass of x, and C G (x) its centraizer. We say that x is an invoution if it has order 2. If u Aut(G), then G u denotes the subgroup of points fixed by u. The trivia eement of G is denoted by e. The type of a permutation identifies the size and number of disjoint cyces the permutation spits into. For exampe, e S n has type (1 n ) and (1 2) S n has type (1 n 2,2). A decomposition of a rack X is a presentation as a disjoint union of two subracks X = Y Z. A rack is indecomposabe if it admits no decomposition Affine racks of type D. LetAbe a finite abeian group andg Aut(A). We denote by (A,g) the rack with underying set A and rack mutipication x y := g(y) + (id g)(x), x,y A; this is a subrack of the group A g. Any rack isomorphic to some (A,g) is caed affine. Remark 2.1. Notice that (A,g) is indecomposabe if and ony if id g is invertibe. For, assume thatim(id g) A; since x y = g(y)+(id g)(x) = y+(id g)(x y), the decomposition of A in cosets with respect to Im(id g) is a decomposition in the sense of racks. In fact, if Y is any coset, then A Y = Y ; thus, the union of any two different cosets is a decomposabe subrack of A. For the converse, note that for any y,z A, x = (id g) 1 (z g(y)) satisfies z = x y; hence (A,g) is indecomposabe. For instance, consider the cycic group A = Z/n and the automorphism g given by the inversion; the rack (A,g) is denoted D n and caed a dihedra rack. Thus, a famiy (µ i ) i Z/n of distinct eements of a rack X is isomorphic to D n if µ i µ j = µ 2i j for a i,j. Lemma 2.2. If m > 2, then the dihedra rack D 2m is of type D. Proof. Since id g = 2 is not invertibe, X = D 2m = {1,2,3,...,2m} is decomposabe. Indeed, if Y = {1,3,5,...,2m 1}, then X Y Y and therefore X = Y (X\Y) is a decomposition. Let r = 1 Y and s = 2 X\Y. Then r (s (r s)) s, since r (s (r s)) = 2+2m and m > 2. We now consider a generaization of this exampe. Let k,t N, k > 1, t > 2 and consider the affine rack (A,g) where A = (Z/k) t 1 and ( ) t 1 g(a 1,...,a t 1 ) = a i,a 1,...,a t 2. (2) i=1 Note that the case t = 2 corresponds to the dihedra rack. Lemma 2.3. If (t,k) 1, and k > 2 when t = 4, then (A,g) is of type D. Proof. First we show that (A,g) contains at east two cosets with respect to Im(id g). Indeed, consider (1,0,...,0) A. Then (1,0,...,0) / Im(id g), Rev. Un. Mat. Argentina, Vo 51-2

5 TWISTED HOMOGENEOUS RACKS OF TYPE D 5 since otherwise there exists (c 1,...,c t 1 ) such that t 1 (1,0,...,0) = (id g)(c 1,...,c t 1 ) = (c 1 + c i,c 2 c 1,...,c t 1 c t 2 ) which impies that c i = c j and 1 = tc 1 in Z/k, a contradiction. Now, take s = (0,...,0) and r = (1,0,...,0); then (2, 2,2, 1,0,...,0), t 5; r (s (r s)) = (2, 2,2), t = 4; (1, 2), t = 3. This is different from s, by hypothesis. Thus (A,g) is of type D. i=1 3. Genera techniques 3.1. Twisted conjugacy casses. Let G be a finite group and u Aut(G). G acts on itsef by y u x = yxu(y 1 ), x,y G. Let O G,u x be the orbit of x G by this action; we ca it the twisted conjugacy cass of x. Then O G,u x is a rack with operation y u z = yu(zy 1 ), y,z O G,u x. (3) The verification that this is a rack is straightforward, see [AG, Remark 3.8]. For instance, if y = h u x and z = v u x, then y u z = w u x, where w = hxu(h 1 v)x 1. We denote this rack by (G,u). Of course, if u = id, then this is just the rack structure on a conjugacy cass. For fixed G, these racks ony depend on the cass of u Out(G). For, if n G and u = uι n, where ι n is the inner automorphism associated to n, then R : (G,u) (G,u ), R(x) = xu(n 1 ), is an isomorphism of racks. Indeed, if x,y G, then R(x y) = xu(yx 1 n 1 ) = xu(n 1 )u(nyx 1 n 1 ) = R(x) R(y). Lemma 3.1. Let x G with ord(x) = 2m > 4 and u(x) = x 1. If x O G,u then O G,u x is of type D. The hypothesis x O G,u x is equivaent to the existence of y G such that y u x = x 2. Proof. Since x i u x j = x i u(x j i ) = x 2i j, x is a subrack of G isomorphic to D 2m ; hence Lemma 2.2 appies. The foowing consequence of (3) heps the search of twisted homogeneous racks of type D. Remark 3.2. If x G u, then O Gu x is a subrack of O G,u x. Note that, if u is an invoution, then x, O G,u x G u z G : zxz = u(x) and y G : z = u(y 1 )y. (4) If k = 2 and t = 4, then (A,g) is not of type D. Rev. Un. Mat. Argentina, Vo 51-2

6 6 ANDRUSKIEWITSCH, FANTINO, GARCÍA, AND VENDRAMIN 3.2. Twisted homogeneous racks. We fix a finite group L, t N, t > 1, and θ Aut(L). Furthermore, we fix the foowing notation for the rest of this section: G = L t, u Aut(G), u( 1,..., t ) = (θ( t ), 1,..., t 1 ), 1,..., t L. The twisted conjugacy cass of (x 1,...,x t ) L t, equipped with the operation defined in (3), wi be caed a twisted homogeneous rack of cass (L,t,θ) and denoted C (x1,...,x t). It is aso usefu to denote C := C (e,...,e,), L, as wi be seen beow in Proposition 3.3. We first describe the twisted homogeneous racks of cass (L,t,θ). Proposition 3.3. (i) If (x 1,...,x t ) L t and = x t x t 1 x 2 x 1, then C (x1,...,x t) = C. (ii) C = C k iff k O L,θ ; hence C = {(x 1,...,x t ) L t : x t x t 1 x 2 x 1 O L,θ }. (iii) There exists a bijection ϕ between the set of twisted conjugacy casses of L and the set of twisted homogeneous racks of cass (L,t,θ), given by ϕ(o L,θ ) = C. (iv) C = L t 1 O L,θ. Proof. (i). Let u j = (x j...x 1 ) 1 ; then (u 1,u 2,...,u t 1,e) (x 1,...,x t ) = (u 1 x 1,u 2 x 2 u 1 1,...,u t 1x t 1 u 1 t 2,x tu 1 t 1 ) = (e,...,e,x t x t 1 x 2 x 1 ). (ii). Suppose that there exists(a 1,...,a t ) L t such that(a 1,...,a t ) (e,...,e,) = (e,...,e,k). Then a t 1 = a t 2 =... = a 2 = a 1 = θ(a t ), hence k = a t θ(a 1 t ). Conversey, assume that k = bθ(b 1 ); if a t = b, a t 1 = a t 2 =... = a 2 = a 1 = θ(b), then (a 1,...,a t ) (e,...,e,) = (e,...,e,k). The second caim foows at once from the first and (i). Now (iii) is immediate. (iv). Define the map ψ : L t 1 O L,θ C by ψ(b 1,...,b t 1,c) = (b 1,...,b t 1,θ 1 (b t 1 b 1 ) 1 c). It is a we-defined map by (ii) and it is ceary injective. Moreover, by the proof of (ii) we know that if (b 1,...,b t ) C, then b t = θ 1 (b t 1 b 1 ) 1 c for some c O L,θ. Thus (b 1,...,b t ) = ψ(b 1,...,b t 1,c) by definition and ψ is surjective, impying that C = L t 1 O L,θ Twisted homogeneous racks intersecting the diagona. Ceary, hence, by Proposition 3.3 G u = {(a,...,a) : a L θ }, (5) C G u a L θ : a t =. (6) Rev. Un. Mat. Argentina, Vo 51-2

7 TWISTED HOMOGENEOUS RACKS OF TYPE D 7 If this happens, then we have an incusion of racks O Lθ a C, see Remark 3.2. In the particuar case θ = id, if there exists a L such that a t =, we have an incusion of racks O L a C Affine subracks of twisted homogeneous racks. Lemma 3.4. Let L. Assume that there exists x L such that moreover if k = ord(x), then we aso require Then C is of type D. θ(x 1 ) = x; (7) (k, t) 1, and additionay (8) k 2,4, when t = 2 and (9) k > 2, when t = 4. (10) When θ = id, (7) just means that x C L (). Proof. Let X = {(x a1,x a2,...,x at 1,x at t ) : i=1 a i 0 mod k}. Then X is a subrack of C since X C, by Proposition 3.3 (ii), and (x a1,...,x at ) (x b1,...,x bt ) = = (x a1,...,x at )u((x b1,...,x bt )(x a1,...,x at 1 )) = (x a1,...,x at )(θ(x bt at 1 ),...,x bt 1 at 1 ) = (x a1+bt at,...,x at+bt 1 at 1 ). Moreover, et (A,g) be the affine rack considered in Lemma 2.2 or Lemma 2.3. Then by the cacuation above, there exists a rack isomorphism (A,g) ϕ X, (a 1,...,a t 1 ) (x a1,...,x at 1,x t 1 i=1 ai ), which impies that X is of type D, by Lemma 2.2 or Lemma 2.3. Coroary 3.5. Assume that t 6 is even. If L θ, with ord() even, then C is of type D. Proof. Take x = and appy Lemma 3.4. Lemma 3.6. Assume t = 2 and θ = id. Let L. Suppose that there exists x,y,a L such that = xy = yx, x e y, x 2 = e, a C L (x) C L (y), a 4 e, x / a. (11) Then C is of type D. Rev. Un. Mat. Argentina, Vo 51-2

8 8 ANDRUSKIEWITSCH, FANTINO, GARCÍA, AND VENDRAMIN Proof. Let X = {(xa i,a i y) : 0 i < ord(a)} and Y = {(a i,a i ) : 0 i < ord(a)}. Both are subracks of C by Proposition 3.3 (ii) and the fact that a C L (x) C L (y). Since x / a, we have that X Y =. Moreover, X Y is a subrack of C since (xa i,a i y) (a j,a j ) = (a 2i j,a 2i+j ) (a i,a i ) (xa j,a j y) = (xa 2i j,a 2i+j y). Taking r = (x,y) and s = (a,a 1 ) we get that r (s (r s)) = (a 3,a 3 ). Since a 4 e by assumption, it foows that C is of type D. Coroary 3.7. Assume t = 2 and θ = id. Let L. If ord() = 2m with m odd, then C is of type D. and Proof. The eements x = m, y = a = 1 m satisfy ord(y) = m and (11). Denote by D n = x,y x 2 = 1 = y n, xyx = y 1 the dihedra group of 2n eements. Lemma 3.8. Assume t = 2 and θ = id. If there exists ψ : D n L a group monomorphism, with n 3 odd, then C ψ(x) is of type D. Proof. Let z = ψ(y), b 1 = ψ(x) and b j = b 1 z j 1, 2 j n. Then z i b j = b j i and b i z j = b i+j, for a 1 i,j n. Let R = {(z i,z i b j ) 1 i,j n} and S = {(z k b,z k ) 1 k, n}. They are disjoint since otherwise there exist i, k, {1,...,n} such that z i = z k b ; this impies that z i+k = b, which is a contradiction because b is an invoution and z has odd order. Note that (z i,z i b j ) = (z i,b j+i ) and (z k b,z k ) = (b +k,z k ). Hence R and S are subracks of C ψ(x) since (z i,b j+i ) (z k,b +k ) = (z 2i+j k,b k+j ), (b j+i,z i ) (b +k,z k ) = (b k+j,z 2i+j k ). Moreover, T = R S is a decomposabe subrack of C ψ(x) because (z i,b j+i ) (b +k,z k ) = (b j k,z (2i+j k ) ), (b +k,z k ) (z i,b j+i ) = (z (2k+ j i),b i ). If we take r = (1,b 1 ) and s=(b 2,1), then r (s (r s)) = (b 1 b 2 b 1,b 1 b 2 b 1 b 2 ) s. Therefore, C ψ(x) is of type D Twisted homogeneous racks with quasi-rea. Let L and j N. Reca that, or O L, is quasi-rea of type j if j and j O L. We observe that might be quasi-rea of different types. For exampe, if is rea, that is 1 O L, but not an invoution, then it is quasi-rea of type ord() 1. We partiay extend this notion to twisted conjugacy casses. We sha say that L θ is θ-quasi-rea of type j N if j and j O L,θ. Note that if L θ is quasi-rea of type j in L θ, then it is θ-quasi-rea, but the converse is not true. Indeed, if we take L = A 11, θ = ι (1 2) the conjugation by (12), Rev. Un. Mat. Argentina, Vo 51-2

9 TWISTED HOMOGENEOUS RACKS OF TYPE D 9 = (1 2)(3 4)( ) and y = (1 4)(2 3)(5 6 8)(7 10 9), then y θ = 2, but 2 O Aθ 11. Lemma 3.9. Assume that L θ is θ-quasi-rea of type j. If t 3 or t = 2 and ord() 2(1 j), then C is of type D. Proof. Let X = {( a1,..., at ) : Y = {( a1,..., at ) : By definition, X Y =. We have t i=1 a i = j}, t i=1 a i = 1}. ( a1,..., at ) ( b1,..., bt ) = ( a1+bt at, a2+b1 a1,..., at+bt 1 at 1 ). (12) Hence X and Y are subracks of C, X Y Y, Y X X, and X Y is a subrack of C. Take r = (1,...,1, j ) X and s = (1,...,1,) Y. We show that r (s (r s)) s. Assume first that t 4. Then by (12) we have that r (s (r s)) = r (s ( 1 j,1,...,1, j )) = r ( j 1, 1 j,1,...,1,) = ( 1 j, j 1, 1 j,1,...,1, j ) (1,...,1,). If t = 3, then r = (1,1, j ), s = (1,1,) and the computation yieds r (s (r s)) = r (s ( 1 j,1, j )) = r ( j 1, j+1,) = ( 1 j, j 1,) (1,1,). Finay if t = 2, then r = (1, j ), s = (1,) and we have r (s (r s)) = r (s ( 1 j, j )) = r ( j 1, j+2 ) = ( 2j+2, 2j 1 ). Thus, C is of type D if ( 2j+2, 2j 1 ) (1,) which amounts to ord() 2(1 j) Summary. The foowing proposition registers the resuts proved in the preceding subsections which wi be usefu in the seque. Proposition Let L θ. (i) If is quasi-rea of type j, t 3 or t = 2 and ord() 2(1 j), then C is of type D. (ii) If ord() is even and t 6 is even, then C is of type D. (iii) If is an invoution, t is odd and O Lθ is of type D, then so is C. (iv) If t = 4 and there exists x C L θ() with ord(x) = 2m > 2, m N, then C is of type D. (v) If t = 2 and there exists x C L θ() with ord(x) = 2m > 4, m N, then C is of type D. (vi) If is an invoution, t = 2, and there exists ψ : D n L θ a group monomorphism, with n 3 and = ψ(x) for some x D n invoution, then C is of type D. Rev. Un. Mat. Argentina, Vo 51-2

10 10 ANDRUSKIEWITSCH, FANTINO, GARCÍA, AND VENDRAMIN (vii) If (t, L θ ) is divisibe by an odd prime p, then C e is of type D. (viii) If (t, L θ ) is divisibe by p = 2 and t 6, then C e is of type D. (ix) If t = 4 and there exists x L θ with ord(x) = 2m > 2, m N, then C e is of type D. (x) If t = 2, and there exists x L θ with ord(x) = 2m > 4, m N, then C e is of type D. Proof. (i) is Lemma 3.9; (ii) is Coroary 3.5; (iii) foows from the discussion in Subsection 3.3, since (,...,) C. Case (vi) foows by Lemma 3.8. Finay, the remaining cases foow from Lemma 3.4: in cases (vii) and (viii), take x of order p, and in cases (iv), (v), (ix) and (x) the given x. 4. Simpe twisted homogeneous racks We now expore when Proposition 3.10 appies to a simpe twisted homogeneous rack with L an aternating group A n, n 5, or a sporadic group. Remark 4.1. If L is simpe and θ = id, t 4,2 and (t, L ) 1, then C e is of type D. This foows from Proposition 3.10 (vii) and (viii) L = A n, n 5, θ = id. In this subsection we prove Theorem 1.3 in this case by appying the resuts obtained above = e. We now treat the Tabe 1. If t 6 even or t odd with (t,n!) 1, then C e is of type D, by Remark 4.1. Assume that t = 4. If n 6, then C e is of type D, by Proposition 3.10 (ix) with x = (1 2)( ). If n = 5, we do not know if C e is of type D. Assume that t = 2. If n 7, then C e is of type D, by Proposition 3.10 (x) with x = (1 2)(3 4)(5 6 7). If n = 5 or 6, then Proposition 3.10 (x) does not appy because the ony possibe even orders of eements are 2 and 4. Nonetheess, we have checked that C e is not of type D, using GAP an invoution. The type of is (1 r1,2 r2 ), with n = r 1 +2r 2 and r 2 even. Assume that = (i 1 i 2 )(i 3 i 4 ) (i 2r2 1 i 2r2 ). Then By [AFGV1, Thm. 4.1], O L is of type D, except for the foowing cases: (1,2 2 ), n = 5; (1 2,2 2 ), n = 6; (2 4 ), n = 8. In particuar, C is of type D for a t odd, except for the cases above. If t 6 is even, then C is of type D, by Proposition 3.10 (ii). Assume that t = 4. If r 2 4, then C is of type D by Proposition 3.10 (iv) with x = (i 1 i 2 )(i 3 i 5 i 7 i 4 i 6 i 8 ). (13) Suppose that r 2 = 2. If r 1 2, then C is of type D by Proposition 3.10 (iv) with x = (i 1 i 3 i 2 i 4 )(j 1 j 2 ), where j 1, j 2 are fixed by. In the case r 1 = 1, Proposition 3.10 (iv) does not appy because the ony possibe even order of eements is 2. Rev. Un. Mat. Argentina, Vo 51-2

11 TWISTED HOMOGENEOUS RACKS OF TYPE D 11 Assume that t = 2. If r 2 4, then C is of type D by Proposition 3.10 (v) with x as in (13). Suppose that r 2 = 2. If r 1 3, then C is of type D by Proposition 3.10 (v) with x = (i 1 i 2 )(i 3 i 4 )(j 1 j 2 j 3 ), where j 1, j 2, j 3 are fixed by. In the case r 1 = 1 or 2, C yieds of type D, by Lemma 3.8 taking ψ : D 3 x := (1 2)(3 4),(1 2)(3 5) L ord() > 2. It is known that: O L is quasi-rea of type 4, if ord() > 3 is odd, O L is quasi-rea of type ord() 1, if ord() is even or 3. (14) Indeed, if ord() > 3 is odd, then 2 O Sn, hence 4 O L ; whereas if ord() is even or 3, then O L = OSn, see [JL, Prop ] or [FH, Exerc. 3.4], hence is rea. Now, if t > 2 or t = 2 and ord() 4, then C is of type D, by Proposition 3.10 (i). On the other hand, if t = 2 and ord() = 4, then Proposition 3.10 (i) does not appy and we do not know if C is of type D L = A n, n 5, θ id. We may suppose that θ is the inner automorphism associated with an eement σ in S n A n. We choose σ = (1 2); thus, θ() = (1 2)(1 2), for a A n. Hence, A θ n = ( {(1 2)} (S {3,...,n} A {3,...,n} ) ) A{3,...,n} (15) and the order of L θ is (n 2)!. Let A n. Reca that C κ = C if and ony if κ O An,θ, by Proposition 3.3 (ii), and notice that κ O An,θ amounts to κ(1 2) O Sn (1 2), the conjugacy cass of (1 2) ins n. Thus, we wi proceed with our study according to the type(1 s1,2 s2,...,n sn ) of (1 2). Notice that ord((1 2)) is even, since (1 2) S n A n, and ord((1 2)) 4 if and ony if s h 1, for some h 4, or s 2 1 and s 3 1. Now, we want to give a description anaogous to (14) for the case θ id. First, we note that O An,θ A θ n if and ony ifs 1 2 ors 2 1. Indeed, this foows from the form of A θ n described in (15) for the ony if part and checking the possibiities for the if part. Assume thats 1 2 ors 2 1 and et κ O An,θ A θ n. SinceO An,θ = O Sn (1 2) (1 2), it is cear that κ is θ-quasi-rea of type j if and ony if κ j (1 2) O Sn (1 2) and κj κ. (16) If ord((1 2)) 4, then κ(1 2) is quasi-rea of type j in S n with j = ord((1 2)) 1, by (14). Thus, κ j (1 2) = (κ(1 2)) j and κ j (1 2) O Sn (1 2). Hence, κ is θ-quasi-rea of type j, by (16); moreover, κ yieds quasi-rea of type j in A θ n. On the other hand, if ord((1 2)) = 2, then κ O Sn (1 2) is θ-quasi-rea (of type j = 0) if and ony if O An,θ contains e, i. e. O Sn (1 2) is the conjugacy cass of the transpositions in S n. Rev. Un. Mat. Argentina, Vo 51-2

12 12 ANDRUSKIEWITSCH, FANTINO, GARCÍA, AND VENDRAMIN We study now the twisted homogeneous racks (A n,t,θ), n 5 and θ id, for different t s according to the type (1 s1,2 s2,...,n sn ) of (1 2). If s 1 1 and s 2 = 0, then O An,θ A θ n =, and we do not know if C is of type D for any t. From now on we wi assume that s 1 2 or s 2 1 and et κ O An,θ A θ n. Notice that C = C κ and that (1 2) and κ(1 2) have the same type, hence the same order. Moreover, if we denote the type of κ by (1 r1,2 r2,...,n rn ), then r h = s h, for a h, 3 h n; thus, ord(κ) divides ord((1 2)). We wi consider different cases. If s h 1 for h = 3 or 5 h n, then ord((1 2)) > 4 and C is of type D for a t, by Proposition 3.10 (i). Assume that s h = 0 for h = 3 and 5 h n. Suppose that s 4 1; thus ord((1 2)) = 4. If t 3, then C is of type D for a t, by Proposition 3.10 (i); whereas if t = 2, we do not know if C is of type D. Assume that s 4 = 0. Thus, the type of (1 2) is (1 s1,2 s2 ), with s 2 1 odd. Suppose that s 2 = 1; thus, e O An,θ. If t 3, then C = C e is of type D for a t, by Proposition 3.10 (i); whereas if t = 2, n 7 and take κ = e, then C = C e is of type D, by Proposition 3.10 (x) choosing x = (1 2)(3 4)(5 6 7). Suppose that s 2 > 1 odd. Then the type of κ O An,θ A θ n is,2 (1r1 r2 ), with n = r 1 +2r 2 and r 2 2 even, i. e. κ = (i 1 i 2 )(i 3 i 4 ) (i 2r2 1 i 2r2 ). We determine now when O Aθ n κ is of type D. We have two possibiities. (i) Assume that κ fixes 1 and 2. If the type (1 r1 2,2 r2 ) is distinct from (1,2 2 ), (1 2,2 2 ) and (2 4 ), then O Aθ n κ is of type D, by [AFGV1, Thm. 4.1]; otherwise, O Aθ n κ is not of type D. (ii) Assume that κ does not fix 1 nor 2; thus κ = (12)(i 3 i 4 ) (i 2r2 1i 2r2 ). If the type (1 r1,2 r2 1 ) is distinct from (2 3 ) and (1 r1,2), then O Aθ n κ is of type D, by [AFGV1, Thm. 4.1]; otherwise, O Aθ n κ is not of type D. We consider now different vaues of t. Assume that t is odd. If κ fixes 1 and 2 and the type of κ is distinct from (1 3,2 2 ), (1 4,2 2 ) and (1 2,2 4 ), then C is of type D, by (i) above and Proposition 3.10 (iii). On the other hand, if κ does not fix 1 nor 2 and the type of κ is distinct from (2 4 ) and (1 r1,2 2 ), for any r 1, then C is of type D, by (ii) above and Proposition 3.10 (iii). If t 6 even, then C is of type D, by Proposition 3.10 (ii). Assume that t = 4. We wi determine when there exists x A θ n such that ord(x) 4 even and θ(κxκ) = x, i. e. (1 2)κxκ(1 2) = x. If κ fixes 1 and 2, take x = (1 2)(i 1 i 3 i 2 i 4 ). If κ(1) = 2 and κ(2) = 1, take x = (1 2)(i 3 i 5 i 4 i 6 ) when r 2 4 and x = (1 j 1 2 j 2 )(i 3 i 4 ) when r 2 = 2 and r 1 2, j 1, j 2 are fixed by κ. In a these cases, C is of type D, by Lemma 3.4. For the remaining cases we do not know if C is of type D. Assume that t = 2. We wi determine when there exists x A θ n such that ord(x) 6 even and θ(κxκ) = x, i. e. (1 2)κxκ(1 2) = x. If κ fixes 1 and 2, take x = (1 2)(i 3 i 5 i 7 i 4 i 6 i 8 ) when r 2 4 and x = (1 i 1 i 3 2 i 2 i 4 )(j 1 j 2 ) when r 2 = 2 and r 1 4, j 1, j 2 are fixed by κ. If κ(1) = 2 and κ(2) = 1, Rev. Un. Mat. Argentina, Vo 51-2

13 TWISTED HOMOGENEOUS RACKS OF TYPE D 13 take x = (1 2)(i 3 i 5 i 7 i 4 i 6 i 8 ) when r 2 4 and x = (1 2)(i 3 i 4 )(j 1 j 2 j 3 ) when r 2 = 2 and r 1 3, j 1, j 2, j 3 are fixed by κ. Therefore the cases where C is not known to be of type D are the foowing (a) (1 2) of type (1 s1,2 s2,...,n sn ), s 1 1 and s 2 = 0 for any t; (b) (1 2) of type (1 s1,2 s2,4 s4 ), s 1 2 or s 2 1, s 2 + s 4 odd, s 4 1 and t = 2; and those in Tabe 4. Finay, from (a), (b) and Tabe 4 we obtain Tabe 2. Tabe 4. n Type of t 6 (1 2,2 2 ) invoution 2 7 (1 3,2 2 ) fixing 1 and 2 2, odd 8 (1 4,2 2 ) odd 10 (1 2,2 4 ) odd 5 (1,2 2 ) invoution 2, 4 6 (1 2,2 2 ) permuting 1 and (2 4 ) odd 4.3. L = sporadic group, θ = id. In this subsection we prove Theorem = e. If (t, L ) 1, with t odd or t 6 even, then C e is of type D; see Tabe 5 for the prime numbers dividing the order of a sporadic group. In particuar, if t 6 even, then C e is of type D since L is even. If t = 2 or t = 4, then C e is of type D, by Proposition 3.10 (ix) and (x), since there aways exists an eement x L of order an invoution. By [AFGV3, Thm. II], if is an invoution then, O L is of type D, except for the cases isted in Tabe 6; in particuar, C is of type D for a t odd, except for these cases. If t 6 is even, then C is of type D, by Proposition 3.10 (ii). If t = 2 or 4, then C is of type D by Proposition 3.10 (iv) and (v), since there aways exists x C L () with ord(x) > 4 even. To see this we use [BR, Bo, Br, GAP, Iv, W, W + ] ord() > 2. The cass O L is rea or quasi-rea, except the casses 6B, 6C of the Suzuki group Suz. Assume that does not beong to the cass 6B or 6C of the Suzuki group Suz. If t 3 or t = 2 and ord() 4, then C is of type D, by Proposition 3.10 (i). Rev. Un. Mat. Argentina, Vo 51-2

14 14 ANDRUSKIEWITSCH, FANTINO, GARCÍA, AND VENDRAMIN Tabe 5. Prime divisors of orders of sporadic groups. L Prime divisors L Prime divisors M 11, M 12 2, 3, 5, 11 Co 1 2, 3, 5, 7, 11, 13, 23 M 22, HS, 2, 3, 5, 7, 11 J 1, 2, 3, 5, 7, 11, 19 McL HN M 23, M 24, 2, 3, 5, 7, 11, 23 O N 2, 3, 5, 7, 11, 19, 31 Co 2, Co 3 J 2 2, 3, 5, 7 J 3 2, 3, 5, 17, 19 Suz, Fi 22 2, 3, 5, 7, 11, 13 Ru 2, 3, 5, 7, 13, 29 T 2, 3, 5, 13 Fi 23 2, 3, 5, 7, 11, 13, 17, 23 He 2, 3, 5, 7, 17 Fi 24 2, 3, 5, 7, 11, 13, 17, 23, 29 Th 2, 3, 5, 7, 13, 19, 31 B 2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 47 J 4 2, 3, 5, 7, 11, 23, 29, 31, 37, 43 M 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 Ly 2, 3, 5, 7, 11, 31, 37, 67 Tabe 6. Casses of invoutions not known of type D; those which are NOT of type D appear in bod. G Casses G Casses G Casses J 2 2A Fi 22 2A Co 2 2A B 2A, 2C Fi 23 2A T 2A Suppose that beongs to the cass 6B or 6C of the Suzuki group Suz. If t 6 even, then C is of type D by Proposition 3.10 (ii); whereas if t = 2, then C is of type D by Coroary L = sporadic, θ id. The sporadic groups with non-trivia outer automorphism group are M 12, M 22, J 2, Suz, HS, McL, He, Fi 22, Fi 24, O N, J 3, T and HN. For these groups the outer automorphism group is Z/2 in a cases. In Tabe 7 we give the orders of L θ when L HN; we cannot determine the order of HN θ with our computationa resources. We wi assume L HN. Tabe 7. Orders of L θ. L L θ L L θ L L θ M M J Suz HS M cl 7920 He 7560 Fi Fi T 96 O N J Rev. Un. Mat. Argentina, Vo 51-2

15 TWISTED HOMOGENEOUS RACKS OF TYPE D 15 We describe the case = e. The remaining cases wi be treated in a separate paper. If t 6 even or t 3 odd and (t, L θ ) 1, then C e is of type D, by Proposition 3.10 (vii) and (viii). In particuar, if t 6 even, then C e is of type D since L θ is even. Assume that t = 2 or t = 4. We have checked with GAP that there exists x L θ such that ord(x) > 4 even. Then C e is of type D, by Proposition 3.10 (ix) and (x). References [AFGV1] N. Andruskiewitsch, F. Fantino, M. Graña and L. Vendramin, Finite-dimensiona pointed Hopf agebras with aternating groups are trivia, Ann. Mat. Pura App. (4), doi: /s , in press. 1, 10, 12 [AFGV2], Pointed Hopf agebras over some sporadic simpe groups, C. R. Math. Acad. Sci. Paris 348 (2010) pp [AFGV3], Pointed Hopf agebras over the sporadic groups, J. Agebra, doi: /j.jagebra , J. Agebra (2011), [AG] [AHS] [AS] [BR] N. Andruskiewitsch and M. Graña, From racks to pointed Hopf agebras, Adv. Math. 178 (2003), , 2, 5 N. Andruskiewitsch, I. Heckenberger and H.-J. Schneider, The Nichos agebra of a semisimpe Yetter-Drinfed modue, Amer. J. Math (2010), , preprint arxiv: N. Andruskiewitsch and H.-J. Schneider, Pointed Hopf Agebras, in New directions in Hopf agebras, 1 68, Math. Sci. Res. Inst. Pub. 43, Cambridge Univ. Press, C. Bates and P. Rowey, Invoutions in Conway s argest simpe group, LMS J. Comput. Math. 7 (2004), [Bo] J. van Bon, On distance-transitive graphs and invoutions, Graphs Combin. 7 4 (1991), [Br] T. Breuer, The GAP Character Tabe Library, Version 1.2 (unpubished); Thomas.Breuer/ctbib/ 13 [FH] W. Futon and J. Harris, Representation theory, Springer-Verag, New York [GAP] The GAP Group, GAP Groups, Agorithms, and Programming, Version ; 2008, ( [HS] I. Heckenberger and H.-J. Schneider, Root systems and Wey groupoids for semisimpe Nichos agebras. Proc. London Math. Soc (2010), , doi: /pms/pdq [Iv] [JL] A. Ivanov, The fourth Janko group, Oxford Mathematica Monographs, The Carendon Press Oxford University Press, Oxford, A. James and M. Liebeck, Representations and characters of groups, Cambridge University Press, Cambridge [J] D. Joyce, Simpe quandes, J. Agebra 79 2 (1982), Rev. Un. Mat. Argentina, Vo 51-2

16 16 ANDRUSKIEWITSCH, FANTINO, GARCÍA, AND VENDRAMIN [W] R. A. Wison, Some subgroups of the Baby Monster, Invent. Math (1987), [W + ] R. A. Wison, R. A. Parker, S. Nickerson, J. N. Bray and T. Breuer, Atas- Rep, A GAP Interface to the ATLAS of Group Representations, Version 1.4, Thomas.Breuer/atasrep, 2007, Refereed GAP package. 13 N. Andruskiewitsch, F. Fantino and G. A. García Facutad de Matemática, Astronomía y Física, Universidad Naciona de Córdoba. CIEM CONICET. Medina Aende s/n (5000) Ciudad Universitaria, Córdoba, Argentina (andrus, fantino, ggarcia)@famaf.unc.edu.ar F. Fantino and G. A. García Facutad de Ciencias Exactas, Físicas y Naturaes, Universidad Naciona de Córdoba. Veez Sarsfied 1611 (5000) Ciudad Universitaria, Córdoba, Argentina L. Vendramin Departamento de Matemática FCEyN, Universidad de Buenos Aires, Pab. I Ciudad Universitaria (1428) Buenos Aires Argentina and Instituto de Ciencias, Universidad de Gra. Sarmiento, J.M. Gutierrez 1150, Los Povorines (1653), Buenos Aires Argentina vendramin@dm.uba.ar Recibido: 5 de mayo de 2010 Aceptado: 4 de diciembre de 2010 Rev. Un. Mat. Argentina, Vo 51-2