Transcript On Isomorphism Testing of Groups with Normal Hall Subgroup

ON ISOMORPHISM TESTING OF
GROUPS WITH NORMAL HALL
SUBGROUPS
Youming 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…

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EXAMPLE
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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.
GI
GpI
Best known algorithm
Self-reducibility
(search to decision)
~
exp( O ( n ))
YES
n
log n  O ( 1 )
?
[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)
 (log
 # of groups can be as large as n



2
n)
Current bottleneck: p-groups ([Wilson] made
effort to understand structure of p-groups).
Effort to formalize this bottleneck: BCGQ.
OUR RESULTS
REVIEW OF GROUP-THEORETIC NOTIONS
Given a group G.
 Order of a group,
subgroup, cosets
 Normal subgroup,
quotient group
 Direct product
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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.

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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

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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

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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.
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REP. THEORY OF FINITE GROUPS IN TCS.
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Fourier analysis of boolean functions:
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Representation theory of F2 over complex number.
 Fourier basis: irreducible representations.
 Fourier transform of boolean function: irreducible
reps form a orthonormal basis of class functions.

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[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.
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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)
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OUR RESULT: A CONCRETE RESULT
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Efficient isomorphism testing of Abelian ⋊ Elem.
Abelian, (|A|, |E|)=1.
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# 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.
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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.
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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 – SchurZassenhaus theorem.

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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.
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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)).
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[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
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Abelian case: group cohomology.
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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)) ◦ ψ
which 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 Fql 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.)