permutation group algorithms - elteweb.cs.elte.hu/summerschool/2016/egyeb/permutation_ii.pdf ·...

Permutation Group Algorithms 2016 1 / 32

Permutation Group Algorithms

Zoltan Halasi

Eotvos Lorand University

2016

More group theory Permutation Group Algorithms 2016 2 / 32

Going into deeper to the structure of groups


Normal subgroups, quotient groups; Composition factors

Normal subgroup: N C G , if N ≤ G and∀g ∈ G : Ng = gN, ⇐⇒ g−1Ng := g−1ng | n ∈ N = N

N C G , g , h ∈ G ⇒ Ng · Nh = N(gh) (as product ofcomplexes)

Quotient group: G/N := Ng | g ∈ G with multiplicationgiven by Ng · Nh = Ngh. (Unit element: N = N · 1, Inverse(Ng)−1 = Ng−1.)

G is simple if N C G ⇐⇒ N = 1 or N = G

Composition series:

1 = Gk C Gk−1 C . . . C G1 C G0 = G ,

with Gi−1/Gi simple for every 1 ≤ i ≤ k .

The name of the Gi−1/Gi ’s: composition factors

Theorem (Jordan-Holder): The composition factors areuniquely determined for a group up to ordering andisomorphism.


The finite simple groups

Every finite group is built up from finite simple groups

Classification of Finite Simple Groups (CFSG, ∼ 1980)Every finite simple group is isomorphic to one of the following:

Abelian: Zp for p prime;Alternating: Altn for n ≥ 5;Groups of Lie type: Related to simple Lie algebras

Chevalley groups:An(q), Bn(q), Cn(q), Dn(q)E6(q), E7(q), E8(q), F4(q), G2(q)Steinberg groups:2An(q2), 2Dn(q2), 2E6(q2), 3D4(q3)Suzuki–Ree group and Tits group:2B2(22n+1), 2F4(22n+1), 2G2(32n+1), 2F4(2)′

One of the 26 sporadic gorups (Named after their discover):

M11,M12,M22,M23,M24, J1, J2, J3, J4,Co1,Co2,Co3,Fi22,

Fi23,Fi ′24,HS ,McL,He,Ru,Suz ,O ′N,HN, Ly ,Th,B,M


Homomorphisms and quotient groups

Homomorphism: ϕ : G 7→ H, product-preserving, i.e.ϕ(xy) = ϕ(x)ϕ(y) for x , y ∈ G .

Image: Im(ϕ) := h ∈ H | ∃g ∈ G : ϕ(g) = h ≤ H;Kernel: ker(ϕ) := n ∈ G |ϕ(n) = 1H C G .

Isomorphism: Bijective homomorphism.ϕ : G 7→ H, Imϕ = H, ker(ϕ) = 1 (Notation: G ' H);

First Isomorphism theoremϕ : G 7→ H is a homomorphism ⇒ Imϕ ' GkerϕGenerators for G with help of homomorphism:Let ϕ : G 7→ H homomorphism, X ⊂ ker(ϕ), 〈X 〉 = ker(ϕ),and Y ⊂ G be any set satisfying 〈ϕ(y) | y ∈ Y 〉 = Im(ϕ)Then X ∪ Y is a set of generators for G .


Blocks and primitivity

In the following if G acts on Ω, g ∈ G and X ⊂ Ω, then letX g := xg | x ∈ X.

Definition

Let G act on Ω in a transitive way

A partition B = B1, . . . ,Bk of Ω is a block system for G iffor every Bi ∈ B and g ∈ G we have Bg

i ∈ BThe elements of B = B1, . . . ,Bk are called blocks

A block system is trivial, if either B = Ω orB = ω1, . . . , ωnThe action on G is primitive if no nontrivial block system forG exists; otherwise the action is called imprimitive

An equivalence relation ∼ given on Ω is called aG -congruence if G preserves this relation, i.e. if α ∼ β thenαg ∼ βg holds for every g ∈ G


Blocks and primitivity

Let G act on Ω in a trnsitive way

A partition B = B1, . . . ,Bk of Ω is a block system if andonly if the relation defined by B (i.e. α ∼ β iff they are in thesame Bi ) is a G -congruence

For each 1 ≤ i , j ≤ k there is a g ∈ G such that Bgi = Bj

As a result, |B1| = |B2| = . . . = |Bk | and k|B1| = |Ω|Let α ∈ Ω be fixed. Then

∆ |α ∈ ∆ ⊆ Ω, ∆ is a block ←→ H |Gα ≤ H ≤ G,∆ −→ G∆ = g ∈ G |∆g = ∆, H −→ αH ⊂ Ω

In particular, the action is primitive ⇐⇒ Gα ≤max

G

If G is a p-group (i.e. |G | = pk is a prime power), thenM ≤

maxG ⇐⇒ M C G and |G : M| = p.

A consequence is that if G is a p-group acting on Ω, then G isprimitive ⇐⇒ |Ω| = p


Induced actions for a block system

Let G ≤ Sym(Ω) transitive and B = B1, . . . ,Bk a block system.

“top action” The elements of G permutes the elements of theset B = B1, . . . ,Bk, which defines an action of G on B ⇒ ahomomorphism G → Sym(B) with kernelg ∈ G |Bg

i = Bi , ∀i“bottom action” The subgroup GBi

permutes the elements ofBi between each other ⇒ The restriction map defines ahomomorphism GBi

→ Sym(Bi )

The whole action of G is somehow built up from these twoactions (a subgroup of the wreath product)

The non-trivial block system B = B1, . . . ,Bk is minimal ifthere is no block B ′1 ⊂ B1 (i.e. part of block system) such that1 < |B ′1| < |B1| (⇐⇒ the action of GB1 on B1 is primitive)It is called maximal if there is no block B ′1 ⊃ B1 with|B1| < |B ′1| < |Ω| (⇐⇒ the action of G on B is primitive)


How to find block systems (Atkinson’s algorithm)

Let G ≤ Sym(Ω) given by a set of generators X and Γ ⊂ Ω.

The algorithm finds the finest block system B = B1, . . . ,Bksuch that Γ ⊂ B1. (Finest means: The |Bi |-s are as small aspossible.)

Applying this for each subset Γ = 1, α with α ∈ Ω \ 1 onecan find all the minimal block systems;

Continuing this with the top action of G in a recursive wayone can find all the block systems for G ;

The algorithm has polynomial running time ⇒ A maximalblock system can be found for G in polynomial time.


Atkinson’s algorithm

Main idea: We maintain an equivalence class ∼ on Ω.

∼ is stored by an array r indexed by the elements of Ω. Ateach step, r [α] ∼ α is a representative of the equivalence classof α such that r [α] = r [β] iff α ∼ β;

As initialisation, let γ1 ∈ Γ be fixed and define r(α) := γ1 forα ∈ Γ and r(α) := α otherwise;

During the algorithm, whenever we find α, β ∈ Ω such thatα ∼ β we apply all generators x ∈ X and modify ∼ to letαx = βx ;

This modification is done by a subroutine usually called“Union-find”, which we describe first.


Psudocode: Union-find

Union-find(r , α, β)

Input: the array r representing ∼ on Ω,

and α, β ∈ Ω representatives;

Result: The equivalence classes of α and β are merged;1 for ω ∈ Ω do2 if r [ω] = β then3 r [ω] := α;


The implementation of Atkinson’s algorithm

As initialisation, we choose γ1 := Γ[1] and define r [α] := γ1

for α ∈ Γ, and r [α] := α otherwise;

We also maintain an other array q (This helps us to decreasethe number of steps the algorithm takes); We initialise q := Γ.

During the algorithm we go through the elements of q; Ifα ∈ q is the next element, then let β := r [α], so α ∼ β. Now,we search for an element x ∈ X satisfyingαx 6∼ (β)x ⇐⇒ r [αx ] 6= r [βx ].

If we find one, then we call Union-find(r , r [αx ], r [βx ]) tomerge the classes of αx and βx , and we append r [βx ] to thelist q (So, we always add an element to q if ends to being therepresentative of its class)

When we run out of the elements of q, then r represents thefinest block system containing Γ in a block.


Pseudocode: Atkinson’s algorithm

Atkinson(X , Γ)

Input: X with 〈X 〉 = G , and Γ ⊂ Ω;

Output: Array r representing the finest block system

containing Γ in a block.

1 r := [ ]; q := [ ];2 for α ∈ Ω do3 if α ∈ Γ then r [α] := Γ[1]; Append α to q;4 else r [α] := α;5 for α ∈ q do6 β := r [α];7 for x ∈ X do8 if r [αx ] 6= r [βx ] then9 Union-find(r , r [αx ], r [βx ]);

10 Append r [βx ] to q;11 return r ;


How to examine a general permutation group?

A permutation group G ≤ Sym(Ω) is given with X s.t.〈X 〉 = G .

We would like to understand the structure of G e.g. computeits composition factors

Divide-and-conquer: We divide a problem into severalsubcases, which we solve independently

For complexity: If we can handle each subcases in polynomialtime, and the number of subcases is also polynomial (in inputlength), then the original problem can also be solved inpolynomial time

Divide-and-conquer for permutation groups:General ⇒ Transitive ⇒ Primitive ⇒ Simple (CFSG)


Reducing to transitive groups

Let G ≤ Sym(Ω) given by X , 〈X 〉 = G1 We apply the orbit algorithm ⇒ Ω1, . . . ,Ωk orbits of G ; If

G is not transitive, then

Restriction maps: ϕi : G → Sym(Ωi );Gi := ϕi (G ) ≤ Sym(Ωi ) is generated by ϕi (X )G is a subdirect product ofG1 × . . .× Gk ≤ Sym(Ω1)× . . .× Sym(Ωk);The composition factors of G can be calculated from thecomposition factors of the Gi -s;We can reduce many problems for the case G ≤ Sym(Ω)transitive.


Reducing to primitive groups

2 For G ≤ Sym(Ω) transitive, we use e.g. Atkinson’s algorithmfor every pair Γ := 1, α, until we find a minimal blocksystem B = B1, . . . ,Bk for G ; (or we get G is primitive)If we found a non-trivial B, then

The top action: Φ : G → Sym(B) can be handled, whereΦ(G ) ≤ Sym(B) is generated by Φ(X );For each i , a set of generators for the setwise stabiliserHi = GBi can be calculated by the orbit-stabiliser algorithmused for Φ(G ) ≤ Sym(B);The action of Hi can be restricted to Bi . Thus, we get Hi is aprimitive permutation group acting on Bi ;G can be viewed as a subgroup of the wreath productH1 o Φ(G );One can continue this process to Φ(G ) ≤ Sym(B) in arecursive way to find primitive components for G ;In that way, many questions can be reduced for primitivegroups.


The O’Nan–Scott theorem

3 The primitive permutation groups are classified.

Theorem (O’Nan–Scott)

Let G ≤ Sym(Ω) be a primitive permutation group

The socle of G : The subgroup generated by all the minimal normalsubgroups of G;

If G ≤ Sym(Ω) is primitive, then soc(G ) ' Sk for some S simplegroup;

If S ' Cp is Abelian, then G is affine; i.e. Ω = soc(G ) = Fkp is avector space, soc(G ) acts on itself regularly (by translations); andthe point stabiliser G0 ≤ GL(k , q) is a matrix group ⇒ other field ofCGT;

Otherwise, there are several possibilities for the action (diagonal,product action, twisted wreath product, almost simple) which aredescribed;

Then one can use CFSG to find the possible values of S.

Isomorphism of Graphs of Bounded Valence Permutation Group Algorithms 2016 18 / 32

Isomorphism of Graphs of Bounded Valence


The graph isomorphism problem

There was a major breakthrough in graph theory at the end of thelast year:

Theorem (Babai, Nov 2015)

Let X1,X2 be two graphs on n vertices. Then there is an algorithmwhich decides whether the two graphs are isomorphic inquasipolynomial time, i.e. the algorithm has running time nlogc n

for some constant c > 0.

In what follows, we present a result of Luks from 1982, which wasalso a starting point of Babai’s work.


Notation and the result of Luks

Let X be a (finite, undirected, simple) graph.

V (X ) = set of vertices of X ; E (X ) = set of edges of X ;

If a, b ∈ V (X ) are adjacent, write (a, b) ∈ E (X );

The degree of a ∈ V (X ) isd(a) := |b ∈ V (X ) | (a, b) ∈ E (X )|;The valence of X is v(X ) := maxd(a) | a ∈ V (X ).For two graphs X1,X2 a graph isomorphism ϕ : X1 → X2 is abijective map ϕ : V (X1)→ V (X2) such that∀a, b ∈ V (X1) : (a, b) ∈ E (X1) ⇐⇒ (ϕ(a), ϕ(b)) ∈ E (X2);

For a graph X let Aut(X ) := All ϕ : X → X isomorphisms.

Theorem (Luks, (1982))

Let X1,X2 be graphs with |X1| = |X2| = n and v(X1) = v(X2) ≤ tbe bounded. Then the isomorphism of X1 and X2 can be decidedin polynomial-time, i.e. which has running time O(nc) for someconstant c = c(t).


Reducibility to group-theoretic problems

Let X1,X2 be graphs. Decide whether X1 ' X2, i.e. there exists agraph isomorphism ϕ : X1 7→ X2?

We can assume that X1 and X2 are connected. Possible waysto reduce the problem for connected graphs:

Connect each vertices of Xi with a newly added vertex ai ;(Problem: the valency increases)Check isomorphism for each pair (C1,C2) of connectedcomponents C1 of X1 and C2 of X2.Then search for a perfect matching in a bipartite graph.

Let X1,X2 be connected graphs, and form X = X1 ∪ X2

(disjoint union). Equivalent:X1 ' X2;∃g ∈ Aut(X ) such that g(X1) = X2, g(X2) = X1;For any set of generators S ⊂ Aut(X ), ∃g ∈ S such thatg(X1) = X2, g(X2) = X1.

Graph isomorphism can be reduced in polynomial-time to theproblem of finding a set of generators for Aut(X ) of a graphX .


Reducibility to group-theoretic problems

Graph automorphisms with fixed edge. Let X be a graph,e = (a, b) ∈ E (X ) and

Aute(X ) = ϕ ∈ Aut(X ) |ϕ(e) = e ⇐⇒ ϕ(a, b) = a, b.

Graph isomorphism is can be reduced in polynomial-time tothe problem of finding generators for Aute(X ).

Let X1,X2 be (connected) graphs, fix an edge e1 ∈ E (X1).For any e2 ∈ E (X2) define a new graph X :

Take the disjoint union X1 ∪ X2;Insert new vertices v1 on e1 and v2 on e2;Connect v1 and v2 with a new edge e.

Find a set of generators S for Aute(X ) in time O(nc).∃ ϕ : X1 → X2 isomorphism s.t. ϕ(e1) = e2

⇐⇒ ∃g ∈ Aute(X ), g(v1) = v2

⇐⇒ ∃s ∈ S , g(v1) = v2.Doing this for every edge e2 ∈ E (X2), X1 ' X2 can be decidedin time |E (X2)| · O(nc) ≤

(n2

)· O(nc) = O(nc+2).


Reducibility to the Color Automorphism Problem

Definition (Color Automorphism Problem)

Let A be a colored set (i.e. a set and a coloring χ : A 7→ colors)and a permutation group G ≤ Sym(A) given by a set ofgenerators. Find a set of generators for the subgroupg ∈ G |χ(ag ) = χ(a), ∀a ∈ A of color preserving maps.

If X is a graph with |V (X )| = n, then finding generators forAut(X ) is polynomially reducible to the Color AutomorphismProblem for a pair (A,G ) where A is a suitable colored set,G ≤ Sym(A) and G ' Sn.

Let A := a, b | a, b ∈ V (X ).G = Sym(V (X )) ' Sn acts on A by a, bg := ag , bg.Define coloring χ : A→ 0, 1 by χ(a, b) = 0, if(a, b) /∈ E (X ) and χ(a, b) = 1, if (a, b) ∈ E (X )

g ∈ G preserves χ ⇐⇒ g ∈ Aut(X ).



Definition (The class of groups Γk)

For a k > 0 integer, the group G ∈ Γk if every composition factorof G is a subgroup of Sk .

If N C G , then G ∈ Γk ⇐⇒ N,G/N ∈ Γk ;

G ∈ Γk and H ≤ G ⇒ H ∈ Γk .

Lemma

If X is a graph with valence ≤ t, then the problem of finding a setof generators for Aute(X ) for an e ∈ E (X ) is polynomial-timereducible to the Color Automorphism Problem for groups in Γt−1.



Define subgraphs Xr ⊂ X , r = 1, . . . , n byXr =

⋃all pathes of length ≤ r through e.

Xr is not an induced subgraph! If a, b ∈ V (Xr ) \ V (Xr−1) and(a, b) ∈ E (X ), then (a, b) ∈ E (Xr+1) \ E (Xr ).

g ∈ Aute(Xr+1)⇒ Xr is g -invariant ⇒ Restriction mapπr : Aute(Xr+1) 7→ Aute(Xr ) (group homomorphism)

Homomorphism Theorem: Aute(Xr+1)/ ker(πr ) ' Im(πr )

Generators for ker(πr )+ Extension of Generators for Im(πr ) toXr+1 ⇒ Generators for Aute(Xr+1)



“Set of parents”: F := U ⊂ V (Xr ) | 1 ≤ |U| ≤ t“parents function”: f : V (Xr+1) \ V (Xr ) 7→ F ,f (a) := u ∈ V (Xr ) | (a, u) ∈ E (Xr+1)“Siblings:” a, b ∈ V (Xr+1) if f (a) = f (b).Equivalence relation on V (Xr+1)⇒V (Xr+1) = ∆1 ∪ . . . ∪∆s (Equivalence classes of siblings)

v(X ) ≤ t ⇒ |∆i | ≤ t − 1 for each i .

ker(πr ) ' Sym(∆1)× Sym(∆2)× . . .× Sym(∆s)⇒ker(πr ) ∈ Γt−1 and a generator set for ker(πr ) can be chosen.

By induction,

Aute(Xr ) ∈ Γt−1 ⇒ Aute(Xr+1)/ker(πr ) ' Im(πr ) ∈ Γt−1

⇒ Aute(Xr+1) ∈ Γt−1 for every r .



Aute(Xr ) ∈ Γt−1 acts on F ⇒ Aute(Xr ) ≤ Sym(F)

A coloring on F :

Fs = U ∈ F | |f −1(U)| = s for 0 ≤ s ≤ t − 1;F ′ = u, v ∈ F | u, v ∈ V (Xr ), (u, v) ∈ E (Xr+1) \ E (Xr );F0 ∩ F ′, F0 \ F ′, F1 ∩ F ′, F1 \ F ′, . . . ,Ft−1 ∩ F ′,Ft−1 \ F ′is a partition of F ⇒ a coloring on F with 2t colors.

g ∈ Aute(Xr ) is extendable to Xr+1 ⇐⇒ g preserves thiscoloring

A polynomial-time algorithm for the Color AutomorphismProblem ⇒ A set of generators for Im(πr ) in polynomial time.

By induction, we can find a set of generators for Aute(Xr )with r = 1, 2, . . . , n⇒ also for Aute(X ) = Aute(Xn).


A polynomial-time solution of the Color AutomorphismProblem for 2-groups

If X1,X2 have valence 3, then the above argument shows that

the question X1?' X2 is reducible in polynomial-time to the

Color Automorphism Problem for G ∈ Γ2 ⇐⇒ G is a2-group;

For a colored set A and a, b ∈ A leta ∼ b ⇐⇒ “a has the same color as b”;

For B ⊆ A and K ⊆ Sym(A) let

CB(K ) := σ ∈ K | bσ ∼ b, ∀ b ∈ B.

We will use the following propertiesCB(K ∪ K ′) = CB(K ) ∪ CB(K ′) for every B ⊆ A, andK ,K ′ ⊂ Sym(A);CB∪B′(K ) = CB′(CB(K )) for every B,B ′ ⊆ A, andK ⊂ Sym(A).


A generalization of the Color Automorphism Problem

We will use divide-and-conquer algorithm to the followinggeneralization

Problem

Let A be a colored set, G ≤ Sym(A) be a 2-group given by a set ofgenerators, B ⊆ A a G-invariant subset and σ ∈ Sym(A). FindCB(Gσ) in polynomial time of |B|.

This is a really generalization of the Color AutomorphismProblem (take B = A, σ = 1)

If CB(Gσ) 6= ∅, then CB(Gσ) = CB(G )σ0 for anyσ0 ∈ CB(Gσ);

Input: (B, σ,X ), where 〈X 〉 = G ;Output: (σ0,X0), where σ0 ∈ CB(Gσ); 〈X0〉 = CB(G ).


By assumption, G acts on B. First we check, whether this actionis transitive.

If not, then we can find a proper disjoint union B = B1 ∪ B2

with B1,B2 are G -invariant;

We have CB(Gσ) = CB2(CB1(Gσ));

Recursive call:

First apply the algorithm to input (B1, σ,X ) to get σ1,X1 suchthat G1 := CB1 (G ) = 〈X1〉 and CB1 (Gσ) = G1σ1;Then apply the algorithm to input (B2, σ1,X1) to get σ2,X2

such that CB(Gσ) = CB2 (G1σ1) = 〈X2〉σ2;

If the action of G is transitive on B, then we can find a system ofmaximal blocks for G .

Since G is a 2-group, every such block system is of the formB = B1,B2 with |B1| = |B2| = |B|/2;

Let H := GB1 = GB2 and τ ∈ G \ H; Then we can find such aτ and a set of generators for H (By using the orbit-stabilizeralgorithm);


Then G = H ∪ Hτ . Furthermore,

CB(Gσ) = CB(Hσ ∪ Hτσ) = CB(Hσ) ∪ CB(H(τσ))

= CB2(CB1(Hσ)) ∪ CB2(CB1(H(τσ)).

This means that the Problem for input (B, σ,X ) can bereduced in polynomial time to four similar problems on setsB1 and on B2 with |B1| = |B2| = |B|/2.

Finally, if B = b, then CB(Gσ) is either Gσ or ∅ (it can bedecided in constant time)

One can prove that by induction that the total time this recursivealgorithm needs is still polynomial in |B|.


Thank you for your attention!

permutation group algorithms - elteweb.cs.elte.hu/summerschool/2016/egyeb/permutation_ii.pdf ·...

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