ON ISOMORPHISM TESTING OF GROUPS WITH NORMAL HALL SUBGROUPSYouming Qiao

Tsinghua University

Joint work with Jayalal Sarma, Bangsheng Tang

OUTLINE

Problem statement Interest from complexity-theoretic perspective Previous work

Our result Group-theoretic prerequisite Strategy and measure for progress Results: a framework, a rep-theoretic problem,

and a concrete result Some proofs in somewhat detail

Finding complement Taunt’s theorem Reduction to linear code equivalence problem

PROBLEM STATEMENT

GROUP ISOMORPHISM

Groups: mathematical language for symmetry

Group isomorphism: (like all other isomorphism problems) ask whether two groups are the same up to “renaming of elements”

Recall graph isomorphism problem…

R0 R1 R2 S0 S1 S2

R0 R0 R1 R2 S0 S1 S2

R1 R1 R2 R0 S1 S2 S0

R2 R2 R0 R1 S2 S0 S1

S0 S0 S2 S1 R0 R2 R1

S1 S1 S0 S2 R1 R0 R2

S2 S2 S1 S0 R2 R1 R0

EXAMPLE

0 1

0 0 1

1 1 0

0 1

0 1 0

1 0 1

GROUP ISOMORPHISM PROBLEM

Group isomorphism problem: given two groups, whether they are the same up to “renaming of elements” Formally, if there exists an bijection of elements

such that for every g, h such that… Hardness depends on representation:

Presentation Permutation group given as generators Cayley table

GROUP ISOM.: FROM COMPLEXITY THEORETIC PERSPECTIVE

Ladner’s theorem: if NP≠P, there are infinite hierarchies between NPC and P.

Few natural candidates not known to be in P nor NP-complete, let alone the “infinite hierarchy”: Factoring, Graph isomorphism, PIT, Group isomorphism, given as Cayley tables.

GpI ≤ GI, while the inverse direction not known. Are they a possible pair?

GROUP ISOM. AND GRAPH ISOM.

Best known algorithm Self-reducibility(search to decision)

GI YES

GpI ?

))(~

exp( nO

)1(log Onn

[Chattopadhyay, Torán, Wagner 10]

GI can not AC0 reduce to GpI.

A conjecture: GI and GpI are not in P. And, under some complexity-theoretic assumption GI doesn’t reduce to GpI!

WHAT WE KNOW ABOUT GROUP ISOM.

General group isom.: quasi-polynomial. Abelian group isom. in linear time. [Kavitha] Abelian ⋊ Cyclic, (|A|, |C|)=1. [Le Gall]

# of groups in these classes: no(1)

# of groups can be as large as

Current bottleneck: p-groups ([Wilson] made effort to understand structure of p-groups).

Effort to formalize this bottleneck: BCGQ.

)(log2 nn

OUR RESULTS

REVIEW OF GROUP-THEORETIC NOTIONS

Given a group G.Order of a group, subgroup, cosetsNormal subgroup, quotient groupDirect product R0 R1 R2 S0 S1 S2

R0 R0 R1 R2 S0 S1 S2

R1 R1 R2 R0 S1 S2 S0

R2 R2 R0 R1 S2 S0 S1

S0 S0 S2 S1 R0 R2 R1

S1 S1 S0 S2 R1 R0 R2

S2 S2 S1 S0 R2 R1 R0

SEMIDIRECT PRODUCT

Semidirect product, example: dihedral subgroup

Semidirect product: Normal Hall subgroup, Schur-Zassenhaus

theorem Semidirect product in TCS.

It relation with zig-zag product [Alon, Lubotzky, Widgerson]: Given groups A, B, and A ⋊B, for certain choices of generator sets of them, Cayley graph of A ⋊B is zig-zag product of Cayley graphs of A and B.

GENERAL STRATEGY

From existing group class one can form new group class by group products

Given a group of the form K\times K, a natural strategy would be to decompose, test components and pasting solutions back together.

e.g. for direct products: Decomposition: [KN], [Wilson]; Testing components: by assumption; Isomorphism of original: by Remak-Krull-Schmidt.

Can we do the same for semidirect products?

CAVEAT FOR SEMIDIRECT PRODUCT

Decomposition: Do not know how to determine if certain normal

subgroup has a complement; Do not know how to identify a normal subgroup

with a complement. Semidirect product is not unique in general:

recall there is an action associated. (an example)

The above two issues are relative easier for normal Hall subgroup:

Decomposition: Schur-Zassenhaus theorem. Not unique: Taunt’s theorem.

OUR RESULT: A FRAMEWORK

Direct product: decomposition (KN, Wilson), pasting (Remak-Krull-Schmidt theorem)

Semidirect product in Hall case: decomposition (Schur-Zassenhaus theorem), pasting (Taunt’s theorem)

The observation: Schur-Zassenhaus theorem is constructive. Taunt’s theorem applies to normal Hall subgroup.

REVIEW OF REP. THEORY OF FINITE GROUPS

A representation of a group is a homomorphism from an abstract group to a general linear group.

Irreducible representation: building blocks of representations. Decomposing representations: efficiently done. Maschke’s theorem.

Equivalence of representations. Representation of elementary abelian groups.

REP. THEORY OF FINITE GROUPS IN TCS.

Fourier analysis of boolean functions: Representation theory of F2

n over complex number.

Fourier basis: irreducible representations. Fourier transform of boolean function: irreducible

reps form a orthonormal basis of class functions. [Raz, Spieker] On log-rank conjecture:

deciding if two perfect matchings form a Hamiltonian cycle. Alice and Bob get two perfect matchings of a

bipartite graph. Want to decide whether they form a Hamiltonian

cycle.

OUR RESULT: A REP-THEORETIC PROBLEM

Given two representations f and g of G over V, (|G|, |V|)=1, test if there exists φ∈Aut(G), such that f · φ and g are equivalent, in time poly(|G|, |V|)

The above problem is equivalent to test isomorphism of groups with abelian normal Hall subgroups.

STATISTICS OF GROUPS

Number of groups of a given size

Abelian group: H(E, C) H(E, E)

OUR RESULT: A CONCRETE RESULT

Efficient isomorphism testing of Abelian ⋊ Elem. Abelian, (|A|, |E|)=1. # of groups in the class: nΩ(log n)

Note that representation and automorphism group of elem. abelian group are well known.

By reduction to linear code equivalence problem. Given two linear subspaces L, L’ of Fnk, if L and

L’ are same up to permutation of coordinates. GI-hard in general. [Babai] gives a singly exponential time.

SOME PROOFS IN SOMEWHAT DETAIL

OUTLINE (I)

Decompose G=N ⋊ H, given that (|N|, |H|)=1. Compute the normal part, N. Compute the complement part, H – Schur-

Zassenhaus theorem. Formulate a condition of testing isom. of G in

terms of… [Taunt 55] Isom. of the normal parts and the complement

parts. Associated actions of the semidirect products.

Motivates the representation-theoretic problem, when the normal parts are elem. abelian.

OUTLINE (II)

For N elem. abelian, H elem. abelian, reduces to Code Isomorphism problem in singly exp. time.

Give two linear subspaces K and L of Fn, if there exists permutation σ∈Sn, such that K and Lσ are the same subspace, in time exp(O(n)). [Babai 10] gives such an algorithm, solving our

problem. [Le Gall 09] allows us to generalize to N

abelian, H elem. abelian.

THE STRATEGY OF SCHUR-ZASSENHAUS

Abelian case: group cohomology.

Non-abelian case: a recursive algorithm. Base case: abelian; Branch according to whether N is minimal; If not minimal: find the minimal T. Then two

recursive calls w.r.t. K/T=SZ(G/T, N/T) and SZ(K, T)

If minimal: P=Sylow p-subgroup of N. Call SZ(G’, N’) where G’ and N’ are normalizer of G and N.

TAUNT’S THEOREM

G1=N1 ⋊ H1, with action τ: H1 → Aut(N1) G2=N2 ⋊ H2, with action γ: H2 → Aut(N2)(Components should be isomorphic at first

hand.) ψ : N1 → N2 φ : H1 → H2(Isomorphism of large groups w.r.t. small

groups) G1 and G2 are isomorphic if and only if for all h∈H1,

TAUNT’S THEOREM (CONT’D)

τ (h) = ψ−1◦ γ(φ(h)) ◦ ψwhich means that conjugating with ψ, τ and γ ◦ φ are the same for every h. If N1 and N2 are elem. abelian F_p^k, τ and γ ◦ φ are naturally representations over F_p^k.The above condition translates to find an isomorophism φ : H1 → H2 such that τ and γ ◦ φ are equivalent.

CODE EQUIV. PROBLEM

In matrix form: L and M are given as d by l matrices, where row vectors are basis. We would like to know if there are G GL(Fq,d) and P permutation matrix, such that

GLP=M Another way to look at it: consider L and M

are set of vectors in Fqd of size l. Then the above question is whether these two sets are the same up to linear transformation.

REDUCTION TO CODE EQUIV. PROBLEM

We want to understand rep. of Fql over Fpk. Fact 1: irr. rep. of Fq

l over Fpk may not be dim.

1. Fact 2: every vector of Fql over Fpk induces

an irreducible rep., but two vectors may induce the same rep. up to equivalence.

A simple observation fv◦φT = fφ(v). Suppose all irr. reps are of multiplicity 1. After decomposition, we get vector sets

V={v1, …, vk} and W={w1, …, wk}. Thus the problem is to find φ such that Vφ=W.

THANKS Questions please.

(Thanks go to J.L. Alperin, James B. Wilson and Laci Babai for helpful comments and knowledge.)