# # G2G2 Minicourse 1 Exercises, Leonard Soicher, June 2021 # ======================================================= # # # First, make sure that GAP (version at least 4.11.0) is # installed on your computer, together with its included # packages GRAPE and DESIGN fully installed. # # I suggest you add your code to this file as you attempt # the exercises, and correct any errors and update the file # with new answers as you go. # # You can copy and paste your code into a running GAP session # to see the results. # # # # Exercise session 1 # ================== # # # Exercise 1 # ---------- # # Start up GAP and run the code below. This code creates objects # you will be using in the exercises to follow. Make sure you # understand what the code is doing. # # We first set M24 to be the Mathieu group M_{24}, # octad1 to be the octad for M24 containing [1..5], # and octads to be the set of all 759 octads for M24. # n:=24; M24:=MathieuGroup(n); L:=[1..5]; H:=Stabilizer(M24,L,OnSets); orbs:=Orbits(H,[1..n]); for orb in orbs do if Length(orb)=3 then octad1:=Union(L,orb); break; fi; od; octad1; octads:=Set(Orbit(M24,octad1,OnSets));; Length(octads); # # We next construct the setwise stabilizer in M24 of [23,24], # determine the kernel and image of its action on [1..22], and # assign this image to autM22 (since this image is isomorphic # to the automorphism group of the Mathieu group M_{22}). # pair:=[23,24]; pair_stab:=Stabilizer(M24,pair,OnSets); hom:=ActionHomomorphism(pair_stab,[1..22],OnPoints);; Kernel(hom); autM22:=Image(hom); Size(autM22); # # Now autM22 acts on the set of *hexads*, which is # the set of all octads containing 23 and 24, but with # 23 and 24 removed from each. We compute this set of hexads. # hexads:=[]; for octad in octads do if IsSubset(octad,pair) then AddSet(hexads,Difference(octad,pair)); fi; od; Length(hexads); # # (Note that hexads is the set of blocks of a 3-(22,6,1) design # having points [1..22].) # # # Exercise 2 # ---------- # # Write a function called SymmetricDifference, with parameters # S and T, such that if each of S and T is a set or collection, # then the function returns the symmetric difference of S and T # (which is the set consisting of all the elements that are in # S or T, but not in both). # # Test your function with some well-chosen examples. # # # Exercise 3 # ---------- # # A *dodecad* is the symmetric difference of two octads that # intersect in exactly two points. # # Use a for-loop containing a break statement to # assign to octad2 the first element of the set octads # that intersects octad1 in exactly two points. # # Now determine the symmetric difference of octad1 and octad2, # and assign the result to dodecad1. # # Determine the (setwise) stabilizer in M24 of dodecad1, # and the kernel and image of the action of this stabilizer # on dodecad1. # # The image should be the small Mathieu group M_{12}. # Check this using DisplayCompositionSeries. # # # Exercise 4 # ---------- # # A *sextet* is a partition of [1..24] into subsets of size 4, # such that the union of any two distinct parts is an octad. # # It is known that every 4-subset of [1..24] belongs to a # unique sextet. # # Let sextet1 be the sextet to which [5,10,15,20] belongs. # Determine sextet1 (as a set of sets). # # Determine the orbit of sextet1 under M24 acting on # sets of disjoint sets, but supress the printing of this orbit. # What is the length of the orbit? # # Determine the stabilizer in M24 of sextet1, and # the orders of the kernel and image of the action # of this stabilizer on the set of six parts of the sextet. # Hence, identify the image. # # # # Exercise session 2 # ================== # # Exercise 5 # ---------- # # Load the GRAPE package. # # Construct the complement of the Hoffman-Singleton graph as # the (unique) generalized orbital graph gamma of vertex-degree # 42 for the (unique) primitive permutation group of degree 50 # and order 252000. # # Classify the cliques of gamma of size 15, up to the action # of gamma.group. # # Now use the GRAPE function Graph to construct the Higman-Sims graph # as the graph delta whose vertices are the cliques of size 15 # in gamma, with two such cliques joined by an edge iff the size # of their intersection is 0 or 8. Make use of gamma.group and # its action on the cliques of size 15 of gamma. # # Determine the number of vertices of delta and the vertex-degrees # of delta. # # Determine the automorphism group of delta. # # What is the index of delta.group in the automorphism group of # delta? # # Now use the function NewGroupGraph to assign to delta a copy of # delta whose associated group is the full automorphism group of delta. # # Determine the global parameters, clique number, independence number, # and chromatic number of delta. # # # Exercise 6 # ---------- # # Now you are to implement a different construction for the # Higman-Sims graph, which is the one originally used by # D.G. Higman and C.C. Sims. # # Call this graph to be constructed HSgraph. # # The vertices of HSgraph are 0,1,2,...,22, and the # the hexads of the 3-(22,6,1) design for autM22 constructed # in the first exercise session. # # Construct HSvertices as a set of these 100 vertices. # # # The joining rules for HSgraph are as follows: # # - 0 is joined only to each of the points 1,2,...,22; # # - if i is in [1..22], then i is joined only to 0 and # the hexads to which i belongs; # # - if H is a hexad then H is joined only to the points in H # and the hexads disjoint from H. # # # Implement these joining rules as a boolean function HSadj of # two vertices of HSgraph, returning true iff the vertices # are joined by an edge in HSgraph. # # Clearly autM22 acts as a group of automorphisms of HSgraph, # in its action on the vertices of HSgraph, fixing 0, # acting naturally on [1..22], and acting by OnSets on the hexads. # # Implement this action as a user-defined action HSact in GAP. # # Now construct HSgraph using the GRAPE function Graph # applied to autM22, HSvertices, HSact, and HSadj. # # Check that HSgraph is isomorphic to the graph delta # constructed previously. # # # Exercise 7 # ---------- # # Now load the DESIGN package for GAP. # # Classify the 2-(12,4,3) designs invariant under the cyclic # group generated by (1,2,3,4)(5,6,7,8)(9,10,11,12), such that # every pair of distinct blocks intersect in at most two points. # # Then, for each such design classified, determine the order of # its automorphism group. # # # Exercise 8 # ---------- # # Consider the group M24 and the set octads from the first # exercise session. # # Construct the block design D (in DESIGN package format), # whose points are [1..24] and whose block-list is octads. # # Check that the automorphism group of D is the group M24. # # Look at the online help for the DESIGN package function WittDesign. # Check that D is isomorphic to WittDesign(24). # # Construct one resolution of D invariant under a Sylow # 23-subgroup of M24. # # (Note that this shows that the image of the action OnSets # of M24 on octads is nonsynchronizing.) #