A Library of Nearly Linear-Time Algorithms
Theorem 6.1.13. The algorithm Centralizer of Normal Subgroup in Tran- Tran-sitive Group(G, N) is deterministic and it runs in nearly linear time
6.2. Composition Series
6.2.1. Reduction to the Primitive Case
GivenG= S ≤Sym(),with|| =n, and an SGS relative to a nonredundant base forG, the basic strategy for constructing a composition series ofGis to find anormal subgroup MG and a faithful permutation representation of degree at mostn forG/M. Moreover, either 1 = M = Gor M =1 but the faithful permutation representation ofG/Mis of degree at mostn/2. Applying this procedure recursively forG/M andM, we find acomposition series ofG in at most log|G|logniterations.
IfGis not transitive then letbe the smallest nontrivial orbit ofGand con-struct atransitive constituent homomorphismϕ:G→Sym() and its kernel M :=ker(ϕ). By Section 5.1.3, this construction can be done in nearly linear time.
IfGis transitive then we use the nearly linear-time algorithm of Section 5.5 to check whetherGis primitive. IfGis imprimitive then this algorithm outputs a nontrivial block, and so we can construct a nontrivial block system, the actionϕ:G→Sym() on the blocks, and the kernelM:=ker(ϕ) of this action.
Using the results of Section 5.1.3 again, this construction can be done in nearly linear time.
IfGis primitive then, by the following lemmafrom [Luks, 1987], it is enough to findanyproper normal subgroup ofG.
Lemma 6.2.2. Let G ≤ Sym(), with|| = n, let B =(β1, . . . , βm)be a base for G, and let G=G[1]≥G[2] ≥ · · · ≥G[m+1]=1be the corresponding point stabilizer chain. Suppose that N G, and let i be the index such that
G=G[i]NG[i+1]N . Then |G:G[i+1]N| ≤n and the action of G on the cosets of G[i+1]N has N in its kernel. Moreover, given B and generators for N , this action can be constructed by a deterministic algorithm in nearly linear time.
Proof. We have |G :G[i+1]N| = |G[i]N :G[i+1]N| ≤ |G[i] : G[i+1]|. Since NG[i+1]N,Nacts trivially on theG-cosets ofG[i+1]N.
To show the algorithmic feasibility of the construction of the action on the G-cosets of G[i+1]N, first observe that, by Theorem 5.2.3, the index i can be found in nearly linear time. Let H:=G[i] ∩G[i+1]N. Note that H can be constructed as the pointwise stabilizer of{β1, . . . , βi−1}inG[i+1]N. Since G[i]≥H≥G[i+1]=(G[i])βi, the set:=βiHis ablock of imprimitivity for the action ofG[i]on the cosets ofG[i+1]. Letdenote the block system consisting of theG[i]-images of. The homomorphismψ:G[i] →Sym() defining the action ofG[i]on the block systemcan be obtained in nearly linear time.
We claim that the image ofψ is permutation isomorphic to the action of G[i] on theG-cosets ofG[i+1]N. Indeed, the mapϕ:g→G[i+1]N gis abi-jection between the permutation domains. This mapϕis well defined, since for g,h∈G[i],
g =h⇐⇒gh−1∈ H=G[i]∩G[i+1]N ⇐⇒ gh−1∈G[i+1]N
⇐⇒G[i+1]N g=G[i+1]N h.
Moreover, (ϕ(g))h=ϕ((g)h), soϕinduces apermutation isomorphism.
Since we can construct the permutation action ofG[i]on the cosets ofG[i+1]N andN acts trivially on these cosets, the permutation action of a generating set ofG= G[i],Nis constructed in nearly linear time.
Corollary 6.2.3. Let G≤Sym(), with|| =n, and let N be a maximal nor-mal subgroup of G. Then G/N has a faithful permutation representation of degree at most n.
By the preceding discussion, finding acomposition series is reduced to the following problem:
Given aprimitive groupG≤Sym(),with|| =n, find generators for a proper normal subgroup, or find a faithful representation (6.7)
on at mostn/2 points, or prove thatGis simple.
The fact that the construction of the action ofG on the cosets ofG[i+1]N in Lemma 6.2.2 can be done in nearly linear time was observed in [Beals and Seress, 1992]. That paper also contains a strengthening of the method, when we do not have generators forN. We shall need this extension in Section 6.2.3.
Lemma 6.2.4. Let G≤Sym(), with|| =n, let B=(β1, . . . , βm)be a base for G, and let G = G[1] ≥ G[2] ≥ · · · ≥ G[m+1] =1 be the corresponding point stabilizer chain. Suppose that NG, and suppose that the index i such that G=G[i]NG[i+1]N is known. Then, given an SGS for G relative to B and generators for a subgroup K≥G[i+1]N , the action of G on the cosets of K can be constructed in nearly linear time by a deterministic algorithm, even if we have no generators for N .
Proof. The first part of the algorithm is similar to the one described in Lemma 6.2.2. Namely, we construct the permutation action ofG[i]on the cosets ofK. The only difference from Lemma6.2.2 is that we computeH :=G[i]∩K (instead of the unknownG[i]∩G[i+1]N). Then, as in the proof of Lemma 6.2.2, we compute:=βiHand the block systemconsisting of theG[i]-images of in the fundamental orbitβiG[i]. The same proof as in Lemma 6.2.2, just replacing each reference toG[i+1]NbyK, shows that theG[i]-action on the cosets ofK is permutation isomorphic to theG[i]-action on. Let1:=, 2, . . . , m
denote the blocks in(withm= |G:K| ≤n).
The new part of the algorithm is that we also have to be able to construct the action of arbitrary elements ofGon the cosets ofK. To this end, we compute a shallow Schreier tree data structure forK relative to the baseB, and for each j∈[1,m] we choose apointσj from j. Using theith tree in the shallow Schreier tree data structure ofG, we write the coset representativeshjcarrying βi toσj as words in the strong generators. The set of all hj is a transversal system forGmodK. Note that because of the nearly linear time constraint, we cannot compute the permutationshj explicitly.
Given someg∈G, for eachj ∈[1,m] we want to determine to which coset ofK the permutationhjgbelongs. First, we computeg−1. We claim that after that, for a fixed j, the coset ofhjgcan be found inO(log3|G|) time. Applying this procedure for the generatorsgofG, the action ofGon the cosets ofK is found in nearly linear time.
Recall our convention that strong generating sets are closed for taking in-verses. Hence we can write a wordw of length at most 2 log|G| +1 whose product isg−1h−1j . We siftw as a word (cf. Lemma 5.2.2) through the first
i −1 levels of the Schreier tree data structure of K. Since any element of G = G[i]K can be written in the form gikfor somek∈K andgi∈G[i], this sifting is feasible. By Lemma 5.2.2, the time requirement isO(log3|G|) and the siftees:=g−1h−1j k∈G[i], for somek∈K, is represented as a word of length O(log2|G|). So we can write a word (in terms ofg and the SGS forG,K) representings−1inO(log2|G|) time. Sinceks−1 =hjg, the coset ofhjgcan be found by looking up to which blocklthe pointβis−1belongs.