ON ISOMORPHISM TESTING OF GROUPS WITH NORMAL HALL SUBGROUPS

1. ON ISOMORPHISM TESTING OF GROUPS WITH NORMAL HALL SUBGROUPS Youming Qiao Tsinghua University Joint work with Jayalal Sarma, Bangsheng Tang

2. 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

3. PROBLEM STATEMENT

4. 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…

5. EXAMPLE

6. 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

7. 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?

8. GROUP ISOM. AND GRAPH ISOM. [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!

9. 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.

10. OUR RESULTS

11. REVIEW OF GROUP-THEORETIC NOTIONS Given a group G. • Order of a group, subgroup, cosets • Normal subgroup, quotient group • Direct product

12. 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.

13. 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?

14. 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.

15. 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.

16. 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.

17. REP. THEORY OF FINITE GROUPS IN TCS. • Fourier analysis of boolean functions: • Representation theory of F2n 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.

18. 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.

19. STATISTICS OF GROUPS • Number of groups of a given size • Abelian group: • H(E, C) • H(E, E)

20. 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.

21. SOME PROOFS IN SOMEWHAT DETAIL

22. 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.

23. 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.

24. 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.

25. 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,

26. 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.

27. 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.

28. 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.

29. THANKS  • Questions please. • (Thanks go to J.L. Alperin, James B. Wilson and Laci Babai for helpful comments and knowledge.)