Structure Forests

In many problems dealing with permutation groups, both in theoretical and computational settings, we can use the orbit structure to reduce the original problem to smaller groups. A further reduction is possible by using the imprim-itivity structure of transitive groups. This way, the original problem is reduced to primitive groups, and the strong structural properties of primitive groups can be utilized. In computations, sometimes it is convenient to organize this reduc-tion process by using astructure forest. A structure forestFfor apermutation groupG≤Sym() is aforest of rooted trees where each vertex is asubset of , and the leaves are the one-element subsets of. The elements ofGact as automorphisms ofF, fixing the roots, such that for each nonleaf vertexv of the forest,G_vacts primitively on the children ofv. Thus, in particular, there is exactly one treeT_Oper orbitOin. InT_O, the vertices are blocks of imprim-itivity in the transitiveG-action onOsuch that each level of the tree defines a partition ofO. It is not possible to insert intermediate levels in that tree, with nontrivial branching, and remain consistent with theG-action on the tree.

As asimple example, consider the cyclic groupG = (1,2,3,4)(5,6) ≤ Sym([1,6]). The structure forest consists of two trees. In the first one, the root is the orbit{1,2,3,4}, it has two children{1,3}and{2,4}, and the tree has the leaves{1},{2},{3}, a nd{4}. In the second tree, the root is the orbit{5,6}

and it has two children{5}and{6}. The edges of the forest are defined by set containment.

In general, a groupG ≤Sym() may have many nonisomorphic structure forests. We may construct one by taking a maximal block system on each orbit of G, then taking maximal block systems of the action on these blocks, etc.

Therefore, astructure forest forGcan be computed in nearly linear time by a deterministic algorithm.

Structure forests play an important role in the current fastest deterministic SGS construction for arbitrary input groups (cf. [Babai et al., 1997b]) and in the parallel (NC) handling of permutation groups (cf. [Babai et al., 1987]).

Exercises

5.1. Suppose that a nonredundant base B is given for someG ≤ Sym().

Prove that for H ≤ G andT ⊆ G, the subgroupH ∪T and normal closureH^G can be computed in nearly linear time by a deterministic algorithm.

5.2. LetS be an SGS forGrelative to the base (β1, . . . , βm). LetSi = S∩ G_{(β1,...,βi−1})and letRibe the transversal computed fromSiforG_{(β1,...,βi−1})

modG(β1,...,βi). Consider the elements of Ri as words inSi. For all j ∈ [1,m],r∈ Rj, a ndg∈Sj, let

r g=rmrm−1· · ·r₁,ri∈ Ri (5.4) be the factorization ofr gas a product of coset representatives.

Introduce asymbol xg for each g∈S. Prove that the set of words obtained by rewriting all equations of the form (5.4) in terms of thexg

gives apresentation forG.Hint:Rewrite any word in thexgrepresenting the identity in the spirit of the proof of Lemma4.2.1.

5.3. Give an example of some transitiveG ≤ Sn with minimal base sizeb such that there existsg ∈Gwith|supp(g)| =n/b.Hint:Take a wreath product of cyclic groups.

5.4. (Cauchy) Suppose that some groupG≤Sym() ha storbits. Forg∈G, define fix(g) :=\supp(g). Prove that

g∈G

|fix(g)| =t|G|.

Hint:Count the number of pairs (α,g) withα ∈,g ∈ G, α ∈ fix(g) two different ways.

For the history of this lemma, see [Neumann, 1979].

5.5. [Sims, 1971a] Let (β1, . . . , βm) be abase for G with the correspond-ing point stabilizer chain G=G^[1]≥ · · · ≥G^[m+1]=1, fundamental orbits 1, . . . , m, and transversals R1, . . . ,Rm. Let ¯1, . . . ,¯m

be the fundamental orbits of G relative to the base ¯B=(β1, . . . , βi−1, βi+1, βi, βi+2, . . . , βm). Prove that ¯i+1⊆i. Moreover, forδ ∈ i, letg ∈ Ri be such thatβi^g =δ. Prove thatδ ∈ ¯i+1if and only if β_i^g₊₁⁻¹ ∈i+1. Based on this observation, design an algorithm for comput-ing an SGS relative to ¯B.

5.6. LetG≤Sym() be transitive, and letα, β ∈. Prove that the smallest block of imprimitivity containingαandβ can be obtained as the com-ponent containingαin the graph with vertex setand edge set{α, β}^G (i.e., the edges are theG-images of the unordered pair {α, β}) or a s the strongly connected component containingαin the directed graph with edge set (α, β)^G (i.e., the edges are theG-images of the ordered pair (α, β)).

5.7. Modify the algorithm described in Section 5.5.1 to findallminimal blocks containingαwithin the time bounds given in Theorems 5.5.1 and 5.5.2.

5.8. LetG≤Sym() be transitive. Modify the algorithm in Section 5.5.1 to compute a minimal block containing a given subset⊆by anearly linear-time deterministic algorithm.

5.9. Lett(m,n) denote the worst-case time requirement of at most mFind operations and at mostn−1Unionoperations, and suppose thatm≥n.

Prove that

(i) if neither the collapse rule nor the weighted union rule is used then t(m,n)∈(mn); and

(ii) if only the weighted union rule is used thent(m,n)∈(mlogn).

5.10. LetG≤Sym(). Prove that any structure forest ofGhas less than 2||

vertices.

5.11. Give examples of groupsG ≤ Sym() with exponentially many (as a function of||) structure forests.

The following four exercises lead to the regularity test from [Acciaro and Atkinson, 1992].

5.12. LetG = S ≤Sym() be transitive, and letα∈. Prove thatGacts regularly onif and only ifG_α=G_α^gfor allg∈S.

5.13. LetG = S ≤ Sym() be transitive, and let α, ω ∈ . Construct a list (T[β]|β ∈ ) in the following way: InitializeT[α] :=ω. Build a breadth-first-search treeL that computes the orbitα^G by the algorithm described in Section 2.1.1. Each time an edge−−−→(β, γ) is added toL, because γ =β^sfor some already constructed vertexβofL and somes∈ S, we also defineT[γ] :=T[β]^s.

Prove thatG_α=G_ωif and only ifT[β^g]=T[β]^gfor allβ ∈and for allg∈ S.

5.14. Based on Exercises 5.12 and 5.13, design a deterministic algorithm that tests whether agroupG= S ≤Sym(),with|| =n, acts regularly on, inO(|S|²n) time.

5.15. LetG = S ≤ Sym(),with|| = n, be transitive. Suppose that for someα, β ∈ we haveG_α = G_β, and letbe the smallest block of imprimitivity forGthat containsαandβ. Prove thatG_α =G_δfor each δ∈.

Based on this observation, modify the algorithm in Exercise 5.14 to run inO(|S|nlogn) time.

5.16. LetG= S ≤Sym(),with|| =n, be transitive. Based on Lemma 4.4.2, design a deterministic algorithm that in O((|S| +logn)nlogn) time either decides thatGis not regular, or outputs at most logngenerators forG.

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