Computing in sporadic groups: an application of symmetric generation by Ben Fairbairn

Computing in Sporadic Groups: An Application of Symmetric Generation Ben Fairbairn University of Birmingham Groups St Andrews, University of Bath, August 2nd 2009 Joint with RT Curtis Ben Fairbairn (University of Birmingham) Computing with Symmetric Generation August 2nd 2009 2 / 16 The Basic Idea A group G may contain a generating set T ⊂ G that is symmetric i.e. NG (T ) acts transitively on the elements of T giving a T a symmetric combinatorial structure. Can turn this idea on its head i.e. prescribe a symmetric combinatorial structure for T and ask “What does G look like?” Can be useful for representing elements of G succinctly as a word in the elements of T . Ben Fairbairn (University of Birmingham) Computing with Symmetric Generation August 2nd 2009 3 / 16 Symmetric Generation - ya wha’ ? We write 2 n for a free product of n copies of C2 , the cyclic group of order 2. We write ti for the involution generating the i th copy of C2 . Fix a transitive permutation group N ≤ Sym(n). We can define an action of N on 2 n thusly - for π ∈ N set tiπ := tπ(i) . Using the above action we can define the semi-direct product P := 2 We can P a progenitor. n : N. Ben Fairbairn (University of Birmingham) Computing with Symmetric Generation August 2nd 2009 4 / 16 Symmetric Generation - ya wha’ ? Any element of P = 2 n : N may be expressed as πw where π ∈ N and w ∈ 2 n , a word in the ti s. Given an element πw ∈ P we may factor P by the subgroup πw P . We express this as :N := G . πw We call this a symmetric presentation of G . We call N the control group of G . We call the ti s the symmetric generators of G . (Modulo notational abuse.) 2 n Ben Fairbairn (University of Birmingham) Computing with Symmetric Generation August 2nd 2009 5 / 16 Good Things Come in Small Packages To establish if G is finite we enumerate the double cosets NgN ⊂ G . Since g = πw for some π ∈ N and w ∈ 2 n we have NgN = NπwN = NwN. An adaptation of the celebrated Todd-Coxeter algorithm can be used to enumerate these and John Bray’s coset enumeration program is extremely good at at running this procedure. JN Bray and RT Curtis “Double coset enumeration of symmetrically generated groups” J. Group Theory 7 (2004) 167-185 Ben Fairbairn (University of Birmingham) Computing with Symmetric Generation August 2nd 2009 6 / 16 2 × 2 =? If G is finite then there is a maximum length for w . We say g ∈ G is symmetrically represented if it is expressed as g = πw with π ∈ N and w ∈ 2 n has minimal length. If g := π1 w1 and h := π2 w2 then g , h ∈ G . What is gh? What is g −1 ? Do we have g = h? Use knowledge from the coset enumeration to answer these questions! Ben Fairbairn (University of Birmingham) Computing with Symmetric Generation August 2nd 2009 7 / 16 Example: the Janko group J1 |J1 |=175 560 2 11 : L2 (11) ∼ = J1 (πt1 )5 So every element can be symmetrically represented as πw with π ∈ L2 (11) ≤ Sym(11) and l(w ) ≤ 4. Compare this with the more ‘traditional’ representations: 266 > 72 = 49 > 11 + 4 = 15(> 10 + 3 = 13). RT Curtis and Z Hasan “Symmetric Representation of the Elements of the Janko Group J1 ” J. Symbolic Computation 22 (1996), 201-214 Ben Fairbairn (University of Birmingham) Computing with Symmetric Generation August 2nd 2009 8 / 16 Example: the Janko group J3 :2 |J3 : 2|=100 465 920 2 120 : (L2 (16) : 4) ∼ = J3 : 2 (πt1 )5 So every element can be symmetrically represented as πw with π ∈ L2 (16) : 4 and l(w ) ≤ 3. Compare this with the more ‘traditional’ representations: 6156 > 92 = 81 > (22 + 1) + 3 = 8(> (22 + 1) + 2 = 7). JD Bradley “Symmetric presentations of two sporadic simple groups” PhD thesis, University of Birmingham (2005) Ben Fairbairn (University of Birmingham) Computing with Symmetric Generation August 2nd 2009 9 / 16 Example: the Conway group ·0 ∼ 2˙Co1 = | · 0|=8 315 553 613 086 720 000 JN Bray and RT Curtis “The Leech lattice Λ and the Conway group ·0 revisited” accepted by the transactions of the AMS 24 2 ( 4 ) : M24 ∼ = ·0 (πt)3 Problem: | · 0 : M24 | is a bit big! Too big in fact for us to enumerate the double cosets M24 w M24 . ! Ben Fairbairn (University of Birmingham) Computing with Symmetric Generation August 2nd 2009 10 / 16 Big is Beautiful 2 212 759 O U : M24 ∼ 12 = 2 : M24 O∆U = 1 If we set H := : M24 then we can try and enumerate the double cosets Hw M24 inside ·0 since | · 0 : H| is much smaller than | · 0 : M24 |. John Bray’s program can do this revealing that there are 19 double cosets of the form Hw M24 inside ·0. Indeed every element of ·0 may written in the form π C w where π ∈ M24 , 12 and l(w ) ≤ 4. C ∈2 Problem: can no longer multiply symmetrically represented elements together. Ben Fairbairn (University of Birmingham) Computing with Symmetric Generation August 2nd 2009 11 / 16 Some Crosses to Bear The group ·0 is the group of automorphisms of the celebrated Leech lattice Λ that fix the origin. Can we use the geometry of this lattice to help us? The subgroup 212 : M24 ≤ ·0 is the stabilizer of a certain configuration of vectors in Λ called a cross - the ‘standard cross’ being the set of vectors {±8ei }, the other crosses being the images of this cross under the action of our symmetric generators. Each double coset Hw M24 corresponds to an orbit of crosses under the action of M24 thus: {±8ei } −→ {±8ei } −→ ‡ −→ ‡ H w π Ben Fairbairn (University of Birmingham) Computing with Symmetric Generation August 2nd 2009 12 / 16 Turning the Crank We have identified what each of the 19 orbits of crosses under the action of M24 and found shortest possible words in the symmetric generators sending the standard cross to each of them. This suggests an algorithm for expressing the product of two elements π1 C1 w1 and π2 C2 w2 in the form π3 C3 w3 with l(w ) ≤ 4. 1 2 3 4 5 6 Find the image of the standard cross {±8ei } under the action of π1 C1 w1 . Call this ‡. Find the image of ‡under the action of π2 C2 w2 . Call this ‡ . Find which M24 orbit ‡ belongs to (by inspection). −1 Look-up short word of symmetric generators w3 that sends ‡ back to {±8ei }. Find the Golay codeword C3 by computing the image of (224 ) ∈ Λ −1 under the action of (π1 C1 w1 )(π2 C2 w2 )w3 Find the M24 element π3 by finding the images of enough vectors of −1 the form 8ei under the action of (π1 C1 w1 )(π2 C2 w2 )w3 c3 . Computing with Symmetric Generation August 2nd 2009 13 / 16 Ben Fairbairn (University of Birmingham) Example: the Conway group ·0 ∼ 2˙Co1 = | · 0|=8 315 553 613 086 720 000 24 2 ( 4 ) : M24 ∼ = ·0 (πt)3 So every element can be symmetrically represented as π C w with π ∈ M24 , C ∈ 212 and l(w ) ≤ 4. Compare this with the more ‘traditional’ representations: 196560 > 242 = 576 > 24 + 24 + 4 × 4 = 64(> 23 + 12 + 3 × 4 = 47). RT Curtis and BT Fairbairn “Symmetric Representation of the Elements of the Conway Group ·0” J. Symbolic Computation 44 (2009), 1044-1067 Ben Fairbairn (University of Birmingham) Computing with Symmetric Generation August 2nd 2009 14 / 16 If you liked this... fairbaib@maths.bham.ac.uk Slides are available at: http://bham.academia.edu/BenFairbairn/Talks PLEASE GIVE ME A JOB! Ben Fairbairn (University of Birmingham) Computing with Symmetric Generation August 2nd 2009 15 / 16 Thank you for listening Ben Fairbairn (University of Birmingham) Thanks! August 2nd 2009 16 / 16