A composition tree for a group G can be viewed as a data structure that present a group in terms of its composition factors. The tree is constructed recursively and the data structure facilitates the rewriting of elements of G in terms of different generating sets.
The basic strategy for computing a composition tree of a matrix group employs a combination of a constructive version of Aschbacher's theorem [Asc84] and constructive recognition algorithms for finite simple groups. The basic algorithms are described in [LG01], [O'B06], [O'B11] while some new ideas introduced in [NS06] are incorporated. A detailed account of the entire CompositionTree procedure appears in [BHLGO11].
The algorithm to construct a composition tree proceeds as follows. Given a group G, either:
The group associated with a leaf need not be simple. It may be cyclic or elementary abelian, a soluble or non-abelian simple primitive permutation group, or an absolutely irreducible matrix group that is simple modulo its centre. The decisions on just which groups are treated as a leaves are partly dictated by complexity considerations, and partly based on the quality of available algorithms to process a leaf. For example, we observe no practical advantage flowing from refining a cyclic group to its composition factors.
Once a composition tree for G = < X > has been constructed, then a second list Y of nice generators are stored with G. We call < Y > the nice group. The intrinsic CompositionTree constructs the nice generators Y as SLPs in X. The rewriting algorithm solves rewriting problems on Y and the resulting SLPs can then be rewritten to provide SLPs on X.
The verbose flag SetVerbose("CompositionTree", n) with n=1, ..., 10 may be used to print increasing levels of information on the progress of the functions.
Verify: BoolElt Default: false
Scalar: FldFinElt Default: 1
KernelBatchSize: RngIntElt Default: 5
MandarinBatchSize: RngIntElt Default: 100
MaxHomFinderFails: RngIntElt Default: 1
MaxQuotientOrder: RngIntElt Default: 10^6
FastTensorTest: BoolElt Default: true
MaxBSGSVerifyOrder: RngIntElt Default: 2000
AnalysePermGroups: BoolElt Default: false
KnownLeaf: BoolElt Default: false
NamingElements: RngIntElt Default: 200
UnipotentBatchSize: RngIntElt Default: 100
PresentationKernel: BoolElt Default: true
Given a matrix group G defined over a finite field, this intrinsic constructs a composition tree for G and returns the tree.Verify: If true, then verify correctness of the tree during construction.
KernelBatchSize: The number of normal generators used to construct the kernel of homomorphism.
MandarinBatchSize: The number of random elements used to check correctness of the outcome of Monte-Carlo algorithms.
MaxHomFinderFails: Assume a negative answer after this many failures of certain Monte Carlo algorithms.
AnalysePermGroups: If false, then always treat the permutation group as a leaf, and do not analyse its structure.
NamingElements: The number of random elements used in calls to LieType and RecogniseClassical.
MaxQuotientOrder: A leaf with larger order will not be fully refined to its composition factors.
FastTensorTest: Use only the fast tensor product test.
MaxBSGSVerifyOrder: If RandomSchreier is used on a leaf and it has order less than MaxBSGSVerifyOrder, then Verify the calculation.
PresentationKernel: Use presentations to obtain kernels, where possible.
UnipotentBatchSize: Batch size for the unipotent kernels.
The argument G must be a matrix group over a finite field for which a composition tree datastructure has previously been constructed. The intrinsic determines if correctness of the composition tree can be verified at modest cost using presentations. In effect, the intrinsic determines whether presentations on nice generators are known for all the leaves.
The argument G must be a matrix group over a finite field for which a composition tree datastructure has previously been constructed. The intrinsic verifies the correctness of the composition tree by using it to construct a presentation for G. If G satisfies the presentation, then return true, and the relators of the presentation as SLPs; otherwise return false. The presentation is on the group returned by CompositionTreeNiceGroup(G).
The argument G must be a matrix group over a finite field for which a composition tree datastructure has previously been constructed. The intrinsic returns the nice group for G.
The argument G must be a matrix group over a finite field for which a composition tree datastructure and the associated nice group H has previously been constructed. The intrinsic returns the word group W for H, and the map from W to H.
NonTrivial: BoolElt Default: true
Leaves: BoolElt Default: false
The argument G must be a matrix group over a finite field for which a composition tree datastructure has previously been constructed. This intrinsic displays information about the nodes in the composition tree for G. The tree is traversed in-order. If parameter NonTrivial is true, then only non-trivial nodes will be displayed. If parameter Leaves is true then only leaves will be displayed.
The argument G must be a matrix group over a finite field for which a composition tree datastructure has previously been constructed. This intrinsic returns the coercion map from SLPs in nice generators of G to SLPs in input user generators of G, as well as the SLPs of the nice generators given in terms the user generators.
The argument G must be a matrix group over a finite field for which a composition tree datastructure has previously been constructed. This intrinsic returns the order of G.
The argument G must be a matrix group over a finite field for which a composition tree datastructure has previously been constructed. Given an element g∈G, return true and an SLP for g in the nice generators of G, otherwise return false.
The argument G must be a matrix group over a finite field for which a composition tree datastructure has previously been constructed. This intrinsic returns a change-of-basis matrix that exhibits the Aschbacher reductions of G given by the composition tree.
The argument G must be a matrix group over a finite field for which a composition tree datastructure has previously been constructed. This intrinsic returns a string description of the reduction at the internal node t in the composition tree for G, as well as the image and kernel of this reduction.
The argument G must be a matrix group over a finite field for which a composition tree datastructure has previously been constructed. This intrinsic returns:
- 1.
- A normal series of subgroups 1 = G0 < G1 < ... < Gk = G.
- 2.
- Maps Gi |-> Si, where Si is the standard copy of Gi / Gi - 1, where i ≥1. The kernel of this map is Gi - 1. Observe that Si may be the standard copy plus scalars Z, and the map is then a homomorphism modulo scalars, so that the kernel is (Gi - 1.Z)/Z.
- 3.
- Maps Si |-> Gi.
- 4.
- Maps Si |-> WordGroup(Si).
- 5.
- Boolean flag true or false to indicate if the series is a true composition series.
- 6.
- A sequence of the leaf nodes in the composition tree corresponding to each composition factor. All maps are defined by rules, so Function can be applied on them to avoid built-in membership testing.
The argument G must be a matrix group over a finite field for which a composition tree datastructure has previously been constructed. This intrinsic returns the minimal integer i such that g lies in the ith-term of the normal series returned by CompositionTreeSeries for G.
Given a matrix group G defined over a finite field, this intrinsic returns true if G has a composition tree and false otherwise.
The argument G must be a matrix group over a finite field for which a composition tree datastructure has previously been constructed. This intrinsic removes all data related to the composition tree datastructure for G.
> G := CGOPlus(4, 5^2); > > T := CompositionTree(G); > > DisplayCompTreeNodes (G: Leaves:=true); node = 2 parent = 1 depth = 1 scalar = 1 info = leaf, GrpAb, cyclic group, order 12 fast verify = true ---------- node = 4 parent = 3 depth = 2 scalar = 1 info = leaf, GrpAb, cyclic group, order 4 fast verify = true ---------- node = 7 parent = 6 depth = 4 scalar = 12 info = leaf, GrpAb, cyclic group, order 24 fast verify = true ---------- node = 8 parent = 6 depth = 4 scalar = 2 info = leaf, GrpMat, almost simple, <"A", 1, 25> fast verify = true ---------- node = 9 parent = 5 depth = 3 scalar = 1 info = leaf, GrpMat, almost simple, <"A", 1, 25> fast verify = true ----------
We now verify correctness of the composition tree. In order to show what is going on we illustrate various pieces of the verification process. We begin by setting up the nice group H for G and its associated SLP group; observe that H = G. After checking that verification can be done quickly, we perform the verification.
> H := CompositionTreeNiceGroup(G); > W := CompositionTreeSLPGroup(G); > > CompositionTreeFastVerification(G); true > > f, R := CompositionTreeVerify(G); > #R; 73
At this point we have verified correctness. However, we now explicitly evaluate the relations R on the generators of H. This step has already been performed by CompositionTreeVerify so it is shown here just for demonstration purposes.
> Set(Evaluate(R, [H.i:i in [1..Ngens(H)]])); { [ 1 0 0 0] [ 0 1 0 0] [ 0 0 1 0] [ 0 0 0 1] }
Now that we know that the composition tree is correct, we ask for the order of G.
> CompositionTreeOrder(G); 11681280000
Express the element g of G as a SLP on the generators of the nice group H.
> g := Random(G); > f, w := CompositionTreeElementToWord(G, g); > Evaluate(w, [H.i:i in [1..Ngens(H)]]) eq g; trueRewrite the SLP in terms of the user-supplied generators for G.
> tau := CompositionTreeNiceToUser(G); > tau; Mapping from: GrpSLP: W to SLPGroup(5)Images of elements of W under tau lie in WordGroup(G).
> v := tau(w); > Evaluate (v, [G.i : i in [1..Ngens(G)]]) eq g; trueTest a random element of the generic group for G for membership. (The generic group will be the general linear group GL(4, 52).)
> x := Random(Generic(G)); > f, w := CompositionTreeElementToWord(G, x); > f; false
Finally, we construct a normal series for G and locate a random element within this series.
> CS, _, _, _, flag := CompositionTreeSeries(G); > "Series is composition series? ", flag; Series is composition series? true > "Length is ", #CS; Length is 10 > > g := Random(G); > CompositionTreeFactorNumber(G, g); 10
In this example, we choose a maximal subgroup of the linear group SL(10, 28) and compute its composition tree.
> X := ClassicalMaximals ("L", 10, 2^8); > G := X[1]; > > T := CompositionTree (G); > > DisplayCompTreeNodes (G: Leaves:=true, NonTrivial:=true); node = 6 parent = 5 depth = 5 scalar = 1 info = leaf, GrpAb, cyclic group, order 255 fast verify = true ---------- node = 9 parent = 8 depth = 5 scalar = 1 info = leaf, GrpMat, almost simple, <"A", 8, 256> fast verify = true ---------- node = 13 parent = 12 depth = 3 scalar = 1 info = leaf, GrpPC, abelian group, order 256 fast verify = true ---------- node = 15 parent = 14 depth = 4 scalar = 1 info = leaf, GrpPC, abelian group, order 256 fast verify = true ---------- node = 17 parent = 16 depth = 5 scalar = 1 info = leaf, GrpPC, abelian group, order 256 fast verify = true ---------- node = 19 parent = 18 depth = 6 scalar = 1 info = leaf, GrpPC, abelian group, order 256 fast verify = true ---------- node = 21 parent = 20 depth = 7 scalar = 1 info = leaf, GrpPC, abelian group, order 256 fast verify = true ---------- node = 23 parent = 22 depth = 8 scalar = 1 info = leaf, GrpPC, abelian group, order 256 fast verify = true ---------- node = 25 parent = 24 depth = 9 scalar = 1 info = leaf, GrpPC, abelian group, order 256 fast verify = true ---------- node = 27 parent = 26 depth = 10 scalar = 1 info = leaf, GrpPC, abelian group, order 256 fast verify = true ---------- node = 29 parent = 28 depth = 11 scalar = 1 info = leaf, GrpPC, abelian group, order 256 fast verify = true ----------
We now set up the nice group H for G and its associated SLP group; observe that H = G.
> H := CompositionTreeNiceGroup(G); > "# of generators of H is ", Ngens(H); # of generators of H is 77 > W := CompositionTreeSLPGroup(G);
After checking that correctness of the composition tree can be verified quickly, we perform verification.
> CompositionTreeFastVerification(G); true > f, R := CompositionTreeVerify(G); > #R; 3028
Evaluate the relations on the generators of H.
> Set (Evaluate (R, [H.i:i in [1..Ngens (H)]])); { [1 0 0 0 0 0 0 0 0 0] [0 1 0 0 0 0 0 0 0 0] [0 0 1 0 0 0 0 0 0 0] [0 0 0 1 0 0 0 0 0 0] [0 0 0 0 1 0 0 0 0 0] [0 0 0 0 0 1 0 0 0 0] [0 0 0 0 0 0 1 0 0 0] [0 0 0 0 0 0 0 1 0 0] [0 0 0 0 0 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 1] }
Express the element g of G as a SLP on the generators of the nice group H. Then rewrite the SLP in terms of user generators for G.
> g := Random (G); > f, w := CompositionTreeElementToWord (G, g); > Evaluate (w, [H.i:i in [1..Ngens (H)]]) eq g; true > > tau := CompositionTreeNiceToUser (G); > tau; Mapping from: GrpSLP: W to SLPGroup(4) > > v := tau (w); > Evaluate (v, [G.i : i in [1..Ngens (G)]]) eq g; true
Next test a random element of the generic group of G for membership of G.
> x := Random (Generic (G)); > f, w := CompositionTreeElementToWord (G, x); > f; false
Finally, we construct a normal series for G.
> CS, _, _, _, flag := CompositionTreeSeries (G); > "Series is composition series? ", flag; Series is composition series? true > "Length is ", #CS; Length is 78