Extensions of Symmetric Generating Sets by Ben Fairbairn | Papers by Ben

Extensions of Symmetric Generating Sets Ben Fairbairn - fairbaib@maths.bham.ac.uk 1 School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK Abstract In this paper we review existing methods of extending symmetric generating sets, namely Transitive Extensions and Subset Extensions before introducing a new approach using wreath products. We proceed to give examples of this new construction, most notably for the groups U3 (2r ). Key words Finite Group, Sporadic Groups, Symmetric Generation 1. Introduction The techniques of symmetric generation have produced new constructions and new presentations for a wide variety of finite groups most notably the sporadic simple groups [5]. Symmetric generating sets have been useful in efforts to facilitate computing in large groups. For example, in [6] Curtis and the author represent elements of the Conway group ·0 (a group of order 8315553613086720000) as a string of at most 64 symbols, and typically far fewer, using a symmetric presentation of ·0 due to Bray and the Curtis [2]. This represents a considerable saving compared to representing an element of ·0 as a permutation of 196560 symbols or as a 24×24 matrix (ie as a string of 242 =576 symbols). The main program of [6] makes it possible to multiply together elements of ·0 represented in this manor and thus makes this means of representing the elements of ·0 practical to use. Given the usefulness of applications of this form it is of great interest to seek symmetric generating sets for large groups. Since the techniques of symmetric generation work so well with smaller groups it is natural to seek methods of extending symmetric generating sets for smaller groups to symmetric generating sets for larger groups. Various approaches to this problem have been tried before and are described in Section 3. The methods discussed are not very general and only apply in a very restricted set of circumstances. We present here a new, more general, approach to this problem and give several examples of it at work, investigated as part of the author’s PhD [9]. Throughout this paper we shall use the standard Atlas notation for groups as described in [3]. In Section 2 we shall recall the basic notions of involutory symmetric generation. In Section 3 we shall recall earlier approaches to the problem of extending symmetric generating sets, most notably the idea of ‘Transitive Extensions’ and ‘Subset Extensions’. In Section 4 we proceed to describe our new approach to the problem namely ‘Wreathed Extensions’. 1 This research was partly supported by EPSRC 1 2. Involutory Symmetric Generation Let 2 n denote the free group generated by n involutions. We write {t1 , t2 , . . . , tn } for a set of generators of this free product. A permutation π ∈ Sn induces an automorphism of this free product, π , by permuting its generators namely ˆ ˆ tπ := π −1 ti π = tπ(i) . i Given a group N ≤ Sn we can use this action to form a semi-direct product P:= 2 n : N . When N acts transitively we call P a progenitor. (Note that some of the early papers on symmetric generation insisted that N acts at least 2-transitively.) Elements of P can all be written as a relator of the form πw where π ∈ N and w is a word in the symmetric generators. Consequently any finitely generated subgroup of P may be expressed as H := w1 π1 , . . . , wr πr for some r. In particular, if we factor P by the normal closure of H in P then we shall express the factor group as 2 n: N := G. w1 π1 , . . . , wr πr We say the progenitor P is factored by the relations w1 π1 , . . . , wr πr and that the above is a symmetric presentation of G. Whenever we write a relator wπ we shall tacitly be referring to the relation wπ = id. We call G the target group. Often these relations can be written in a more compact form by simply writing (πw)d for some positive integer d. It is the opinion of the author that no confusion should arise from calling both t ∈ P and its image in G a symmetric generator. Similarly no confusion should arise from calling both N ≤ P and its image in G the control group. Symmetric presentations are readily converted to conventional presentations in terms of generators and relations. Let N = X|R for some generating set X and set of relations R. We write N0 for the stabilizer of the symmetric generator denoted t0 , so that the progenitor has the conventional presentation 2 n : N = X, t|R, t2 , [t, N0 ] . A progenitor factored by the relations w1 π1 , . . . , wr πr can therefore be expressed as 2 n:N = X, t|R, t2 , [t, N0 ], w1 π1 , . . . , wr πr . w1 π1 , . . . , wr πr There are several general lemmata that make symmetric generation ideally suited to the problem of understanding finite simple groups. Lemma 1. Let P = 2 n : N be a progenitor in which the control group, N , is perfect. Then any homomorphic image of P is either perfect or possesses a perfect subgroup of index 2. If w is a word in the symmetric generators of odd length then the factored progenitor 2 n: N πw is perfect. Lemma 2. If N is perfect and primitive, then |P : P |=2 and P = P . Corollary 3. If N is perfect and primitive then any image of P possesses a perfect subgroup of index at most 2. 2 Lemma 4. Every finite simple group is an image of a progenitor of the form 2 n : N. The proofs of these lemmata are given by Curtis in [5]. In addition to the above, there are several general lemmata that, given a progenitor, naturally lead us to relations to factor by providing elementary constructions of a variety of groups. Most notably the following easy lemma proves remarkably powerful in leading to such constructions and in particular has proved enourmously powerful in producing elementary construction of most of the sporadic simple groups [5]. Lemma 5. ti , tj ∩ N ≤ CN (StabN (i, j)) 3. Transitive Extensions and Subset Extensions 3.0.1. Transitive Extensions Let N be a k-transitive permutation group acting on a set X. Recall that a group M is said to be a transitive extension of N if M acts k + 1 transitively on a set X ∪ {α} and the point stabilizer Mα ∼ N . See Dixon and Mortimer [7, Section 7.5] for details. = Transitive extensions may be used to extend symmetric generating sets by extending the corresponding control group. Consider the symmetric presentation G= 2 n:N π1 w1 , . . . , πr wr where πi ∈ N for every i and each wi is a word in the symmetric generators. Naturally the existence of M enables us to extend the progenitor 2 n : N to the progenitor 2 (n+1) : M and thus gives us the possibility of extending the symmetric presentation 2 n: N := G w1 π1 , . . . , wr πr to a homomorphic image of 2 (n+1) : M . Alas, the na¨ approach of simply considering ıve the symmetric presentation 2 (n+1) : M w1 π1 , . . . , wr πr is doomed to failure for the simple reason that the relations will fail to satisfy Lemma 5 applied to the extended progenitor. To extend a homomorphic image of 2 n : N to a homomorphic image of 2 (n+1) : M we consider the factored progenitor P+ = 2 (n+1) :M . (s0 s1 )4 Here any two non-commuting symmetric generators (denoted s0 and s1 ) will generate a dihedral group of order 2k with trivial center if k is odd and a center of order 2 generated by (s0 s1 )k/2 if k is even. We thus factor by the relation (s0 s1 )4 to ensure that ti := (s0 si )2 (i = 0) is an involution in our new progenitor and that ti is in the center of the copy of D8 generated by s0 and si (we can do this since M acts at least 2-transitively). Also we see the ti have no further relations between them in P + and so we can identify a copy 3 of P inside P + . We can thus factor by the appropriate relations to ensure that we have a copy of G as a homomorphic image of P + . Thus we consider the group P+ , π1 w1 , . . . , πr wr where the πi are considered the natural elements of M0 and the wi are written according to the rule ti = (s0 si )2 as defined above. We thus have the group G+ = 2 (n+1) : M . (s0 si )4 , π1 w1 , . . . , πr wr The group G+ is then the transitive extension of G. Several interesting examples of this construction exist and were investigated extensively by Bolt [1]. Perhaps the most striking example of a transitive extension is as follows. Recall that the group L2 (11) acts 2-transitively on 11 points. We can therefore form the progenitor 2 11 : L2 (11). In [4] Curtis proved the symmetric presentation 2 11 : L2 (11) ∼ = J1 (σt0 )5 where σ ∈ L2 (11) is a well chosen permutation. The relation is very naturally motivated using the lemmata given in Section 2 and has been used to explicitly construct J1 both interms of the graph on 266 vertices naturally preserved by the group and in as 7 × 7 matrices over the field F11 . See [5, Section 5.2] for details. The above 2-transitive action of L2 (11) on 11 points can be extended transitively to the 3-transitive action of the Mathieu group M11 on 12 points. Using Soicher’s Coxeter style presentation of the O’Nan group O’N given in the Atlas [3, p.132] (now proved complete) the above symmetric presentation now gives us 2 12 : M11 ∼ O’N:2. = (σ(s0 s∞ )2 )5 , (s∞ s0 )4 , (σ 3 s∞ s3 )5 3.0.2. Subset Extensions Another previously used method of extending a symmetric generating set is as a subset extension. If a k-transitive permutation group N acting on a set of n points X can be extended to a k + 1-transitive permutation group M acting on a set of n + 1 points X ∪ {α} with point stabilizer Mα ∼ N then M will also act transitively on the k + 1 = n element subsets of X ∪ {α}. Any progenitor of the form 2 ( r ) : N for some r ≤ k defined by the action of N on the r element subsets of X can therefore be extended to the n+1 progenitor 2 ( r+1 ) : M defined by the action M on the r + 1 element subsets of X ∪ {α}. Perhaps the most striking example of this construction comes from the large Mathieu groups. The group M22 acts 3-transitively on a set of 22 points that we shall call X. The group will therefore act transitively on the subsets of X of size 2. Again using the lemmata of Section 2 we are naturally lead to the symmetric presentation 22 2 ( 2 ) : M22 ∼ = U6 (2) (σtab )3 where a, b ∈ X and σ ∈ M22 is a well-chosen permutation. The 3-transitive action of M22 on 22 points can be transitively extended to the 4-transitive action of the Mathieu 4 group M23 on 23 points. This allows us to extend the above symmetric presentation to the symmetric presentation 23 2 ( 3 ) : M23 ∼ = Co2 (σtabα )3 where X ∪ {α} is the set naturally acted on by M23 and Co2 denotes the smallest of Conway groups [3, p.154]. This process can be repeated extending the control group to the Mathieu group M24 giving the symmetric presentation 2 ( 4 ) : M24 ∼ = ·0 (σtabαβ )3 where X ∪ {α, β} is the set naturally acted on M24 and ·0 denotes the double cover of the largest Conway group Co1 [3, p.180]. Further examples are given by the author’s symmetric presentations of the finite simply laced Coxeter groups extending the natural action of the symmetric group Sn on n points given in [8]. Both Transitive Extensions and Subset Extensions depend on the action defining the progenitor being highly transitive. Many interesting actions are no more than 1 transitive and most actions have no transitive extensions at all. Consequently the methods of extending transitive actions that we have described in this section cannot be used to provide symmetric generating sets for large groups very often as born out by the sporadic nature of the above examples. In our next section we describe a new approach to extending symmetric generating sets that works much more generally - it can be used to extend any symmetric generating set to a larger symmetric generating set. 4. Wreathed Extensions Let 2 n 24 :N ∼ =G R be a symmetric presentation of the group G defined by some set of relations R. Let M be a transitive permutation group of degree m. We define a wreathed extension of the above symmetric presentation to be a symmetric presentation 2 nm : (N M ) ∼ + =G R+ where the progenitor is defined by the natural action of N M on nm points; R+ is a set of relations containing R and G+ is a group that contains G as a subgroup. As an example of this we show how symmetric generating sets for the unitary groups U3 (2r ) may be obtained by extending symmetric generating sets of the linear group L2 (2r ). In keeping with Atlas notation we write 2r + 1 for the cyclic group of order 2r + 1 and write U3 (2r ) where some authors would write U3 (22r ). Theorem 6. Let r ≥ 2. (i) The group (2r + 1) × L2 (2r ) (and therefore the group L2 (2r )) r is a homomorphic image of the progenitor 2 (2 +1) : (2r + 1). (ii) The group SU3 (2r ) (and therefore the group U3 (2r )) is a homomorphic image of the progenitor 2 ((2r+1)+(2r+1)) : ((2r + 1) 2). Furthermore the symmetric generators 5 in a block of the action defining this progenitor along with the subgroup (2r + 1)2 ≤ (2r + 1) 2 generate a maximal subgroup with structure (2r + 1) × L2 (2r ) that stabilizes a non-isotropic vector in the module naturally acted by SU3 (2r ) (and so any symmetric presentation of SU3 (2r ) as an image of this progenitor may be given as a wreathed extension of a symmetric presentation given in part (i). Proof. We give explicit elements of the group that generate the group and satisfy the relations of the conventional presentation corresponding to the progenitors in question. Let α ∈ F22r be a generator of F× and let V be the natural 3 dimensional F22r module 22r acted on by SU3 (2r ). Consider the following matrices    x=  α2 r −1 0 α2 r 0 0 α2 2r       0 0 1   0 1 0  0 0 −1 0 −1−2(2r −1)   r   r y =  α2 +1 1 α2 +1    1 0 0     t = 1 0 0.   0 0 1 (i) We consider the group generated by the matrices t, x and xn xy for some n. Since each of the matrices x and t has determinant 1, any words in these matrices will generate a subgroup of SL3 (22r ). Now by direct calculation we have that  α2 2r −1−2(2r −1) (2r −1)n α 0 0    r r 2r r r r r  2r  xn xy =  α2 −1−2(2 −1) α(2 −1)n (α2 −1−2(2 −1) − α2 −1 ) α2 −1 α(2 −1)n . 0   r 2r r 0 0 α2 −1 α(2 −1−2(2 −1))n Both this and the matrix t clearly stabilize the 1 dimensional subspace of V spanned by the vector (0,0,1). They thus generate a subgroup of (2r + 1) × L2 (2r ) =: T , the stabilizer r 2r r of this subspace. If n is chosen so that α2 −1 α(2 −1−2(2 −1))n = 1 then xn xy will belong to the derived subgroup of T and since t is an involution it will also belong to the derived subgroup of T since there are no elements of even order in the center of T . We therefore have that xn xy , t ≤ L2 (q). We claim that xn xy , t = L2 (q). We now consider Dickson’s theorem to show that there is no maximal subgroup containing both xn xy and t. r First note that since xn is central and |xn | = |xy | = 2r +1 we have that (xn xy )(2 +1) = n (2r +1) y (2r +1) n y r y (x ) (x ) = id, so x x has order 2 + 1 since no power of x is central. The element xn xy is not contained in a subfield subgroup. Since |xn xy | is a factor of 2r + 1, xn xy is contained in neither a dihedral group of order 2(2r − 1) nor a Froebenious group of structure 2r : (2r − 1). The two elements could only be contained in a dihedral group of order 2(2r + 1) if t normalized the subgroup generated by xn xy . Since xn xy is lower triangular all of its powers are lower triangular, so in particular all elements in the subgroup generated by xn xy are lower triangular. By direct calculation (xn xy )t is upper triangular, so in particular it is not lower triangular, so xn xy , t cannot be dihedral. We thus have xn xy , t = L2 (q). Clearly x commutes with both t and xn xy and so generates a cyclic group commuting with the whole of our L2 (2r ) and since |x| = 2r + 1, t, x, xy ∼ (2r + 1) × L2 (2r ). = Finally, the progenitor in this case corresponds to the conventional presentation 6 x, t|x2 r +1 , t2 and the above matrices clearly satisfy these relations. Consequently (2r + 1) × L2 (2r ) is r an image of the progenitor 2 (2 +1) : (2r + 1) as required. (ii) In this case we have the following conventional presentation for the progenitor x, y, t|x(2 r +1) , y 2 , [x, xy ], t2 , [t, x] . We now observe that the matrices x, y and t given above satisfy the relations of this presentation. We further note that each of the above matrices have determinant 1 and thus certainly generate a subgroup of SL3 (2r ). Furthermore, if P is any of the above matrices then P T AP σ = A where A is the matrix  1 α2 r +1 0    2r +1 α  0 T 1 0   0;  1 P denotes the transpose of the matrix P and P σ denotes the image of P = (pij ) under r r the map pij → p2 . Note that the determinant of A is 1+α2(2 +1) which is non-zero since, ij 2r r r by the definition of α, we have 1 = α2 −1 = (α2 +1 )2 −1 and 2r − 1 is odd. The matrices therefore preserve the non-singular Hermitian bilinear form defined by A and so are contained in the copy of SU3 (2r ) preserving this form. We note further that the matrices x and t each fix the one dimensional subspace spanned by the vector (0,0,1), which is isotropic with respect to the above bilinear form. These matrices are therefore contained in the maximal subgroup of SU3 (2r ) stabilizing the non-isotropic vector (0, 0, 1). (The maximal subgroups of the groups U3 (2r ) were determined by Hartley. See King, [10, p.7], r for details.) Since y clearly maps the vector (0,0,1) to (1, α2 +1 , 0) = (0, 0, 1) these three matrices must altogether generate the whole of the group SU3 (2r ). 2 We remark that this Theorem does not easily extend to characteristics other than 2 - The group SU3 (3) (and therefore the group U3 (3)) is not an image of the progenitor 2 (4+4) : (4 2) despite the stabilizer of of an non-isotropic vector having structure 4˙S4 . We note that in the smaller examples of symmetric presentations whose existence is alluded to by theorem 6, relations completely defining the groups are readily found. In particular we have the following. Lemma 7. Let x and y satisfy the presentation x, y|x5 , y 2 , [x, xy ] ∼ 5 2 then = 2 (5+5) : (5 2) ∼ = U3 (4). (t1 x2 xy )3 , (t1 y)5 The proof of this lemma is simply the observation that the matrices given in theorem 6 satisfy the additional relations and a straightforward (single) coset enumeration. The symmetric generators in a single block satisfy the classical presentation of A5 given by x, t|x5 , t2 , (xt)3 . Finally to illustrate that varied applications of this construction are possible we give examples of this construction in which the control group is not cyclic. 7 Lemma 8. (i) 2 3 : S3 ∼ = S4 (t1 (1, 2))3 (ii) 2 (3+3) : (S3 2) ∼ 2 × L3 (7).2 = (t1 (1, 2))3 , (2, 3)(5, 6)(t1 t4 )4 where S3 2 ∼ (1, 2), (1, 2, 3), (1, 4)(2, 5)(3, 6) . = (Part (i) is a special case of an essentially classical presentation discussed in some detail in [5, Theorem 3.2]. Part (ii) of this lemma is essentially the presentation of L3 (7).2 given in the “Addenda and Corrigenda” section of the Atlas [3, p.xxxiv].) Lemma 9. 2 (4×2) : (2 S4 ) ∼ = 2 × HS:2 (t1 t2 )3 , (t1 π)7 where 2 S4 ∼ (1, 2), (1, 3)(2, 4), (1, 3, 5, 7)(2, 4, 6, 8) and π is the permutation = (1, 3, 8, 15, 16, 14, 9, 2)(4, 10, 12, 11, 13, 7, 6, 5). Here HS denote the sporadic Higman Sims group [3, p.80]. In this case the symmetric generators contained in a block of the action of the control group along with the stabilizer of that block altogether generate a soluble group of order 480. 8 References [1] S.W. Bolt. Some Applications of Symmetric Generation. PhD. thesis. University of Birmingham (2002). [2] J.N. Bray and R.T. Curtis. The Leech Lattice, Λ, and the Conway Group ·0 Revisited. accepted by the Trans. Amer. Math. Soc. [3] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker and R.A. Wilson. An ATLAS of finite groups (Oxford University Press, 1985). [4] R.T. Curtis. Symmetric Presentations II: The Janko Group J1 . J. Lond. Math. Soc. (2) 47 (1993) 294-308. [5] R.T. Curtis. Symmetric Generation of Groups with Applications to many of the Sporadic Finite Simple Groups, Encyclopedia of Mathematics and Its Applications 111 (Cambridge University Press, 2007). [6] R.T. Curtis and B.T. Fairbairn. Symmetric Representation of the Elements of the Conway Group ·0. J. Symbolic Comput. 44 (2009) p.1044-1067. [7] J.D. Dixon and B. Mortimer. Permutation Groups (Springer 1996) [8] B.T. Fairbairn. Symmetric Presentations of Coxeter Groups. submitted [9] B.T. Fairbairn. On the Symmetric Generation of Finite Groups. PhD. thesis. University of Birmingham, 2008. [10] O.H. King. The Subgroup Structure of Finite Classical Groups in Terms of Geometric Configurations. In Survey in Combinatorics, 2005 (ed BS Webb) (Cambridge University Press, 2006). Also available at the time of writing from http://www.staff.ncl.ac.uk/o.h.king/KingBCC05.pdf 9