Probabilistic and Non-Deterministic Computations in Finite Groups

Transcription

1 Probabilistic and Non-Deterministic Computations in Finite Groups A talk at the ICMS Workshop Model-Theoretic Algebra and Algebraic Models of Computation Edinburgh, 4 15 September 2000 Alexandre Borovik 5 September

2 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups Black Box X x y z... random, independent, uniformly distributed elements given as permutations, or matrices in GL n (q),... Format : n 2 log q

3 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups X x y z... We can compare: x = y? multiply, invert: x y, x 1 find orders:... x

4 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups Given a black box group X, Identification problem: Determine the isomorphism type of X with given degree of certainty. Verification problem: Is X isomorphic to the given group G?

5 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups Example n = pq, p and q are primes X = SL 2 (Z/nZ) Determination of the isomorphism type of X as a black box groups amounts to factorisation of n: SL 2 (Z/nZ) SL 2 (F p ) SL 2 (F q )

6 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups Main example X GL N (F q ) simple matrix group X = x 1,..., x k Distribution of orders of elements is specific for every finite simple group and allows to determine X Routines for GAP and MAGMA People working in the area: Babai, Bratus, Celler, Cooperman, Finkelstein, Holt, Leedham-Green, Linton, Kantor, Lubotzky, Murray, Neumann, Niemeyer, O Brien, Pak, Palfy, Praeger, Rees, Seres,...

7 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups Comments 1. Substitute for prime factorisation X GL N (F q ) GL N (F q ) = q N(N 1)/2 (q 1) (q N 1) = r a 1 1 ra m m the finest factorisation one can get x = the least number l = r b 1 1 rb m m, x l = 1.

8 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups Nature of algorithms Verification algorithms are one-sided: Is X G (the target group)? If X contains an element of order not present in G, then X G absolutely

9 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups Example X = SL 2 (2 n ) elements of even order are involutions involutions are conjugate the centralisers and conjugacy classes of involutions: (( )) 1 1 C = C X 0 1 = C = 2 n {( #(involutions ) = X 2 n P (even order) = X 2 n )} 1 X = 1 2 n If n 1, X, for all practical purposes, contains no involution.

10 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups Statistical approach fails for q odd: B n (q) = P Ω 2n+1 (q), C n (q) = P Sp 2n (q), they have virtually the same statistics of orders of elements.

11 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups Why? Let G = G(F q ) be a simple algebraic group regular semisimple elements form an open subset of G statistics of orders of regular semisimple elements is determined by the Dynkin diagram of G, which are the same in the case of groups B n and C n, n 3: BC n, n 2...

12 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups But the conjugacy classes and the structure of centralisers of involutions are determined by the extended Dynkin diagrams which are different: B n, n 3... C n, n 3...

13 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups Centralisers of involutions Assume x n = 1 for all x X, where n = 2 l m, m odd. Miller-Rabin: x x m, (x m ) 2,..., (x m ) 2r 1, 1 i(x) := (x m ) 2r is an involution Brauer: If t is an involution, we have a map ζ : X C X (t) ζ : x { ζ0 (x) = i(t t x ), t t x even ζ 1 (x) = (t t x ) (m+1)/2 x 1, t t x odd ζ = ζ 0 ζ 1

14 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups If c C X (t), ζ 1 (cx) = (tt cx ) (m+1)/2 x 1 c 1 = (tt x ) (m+1)/2 x 1 c 1 = ζ 1 (x) c 1 If x X are uniformly distributed and independent, then ζ 1 (x) are uniformly distributed and independent in C X (t) Hence we have constructed a black box for C X (t) from a black box for X.

15 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups Structural theory of black box groups If X is non-simple, how one can find a normal subgroup? X X = there are involutions t Y X = Y C X (t) C X (t) How one can construct a good black box for the normal closure y X 1,..., yk X?

16 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups Problem: can we construct black boxes for unknown groups? X GL N (F q ), X = x 1,..., x k Given x 1,..., x k, can we construct a black box for X? Yes (Babai) Celler / Leedham-Green / Murray / Niemeyer / O Brian: Product replacement algorithm

17 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups Product replacement algorithm Γ k (G) set of generating k-tuples graph with edges: (g 1,..., g i,..., g k ) (g 1,..., g ±1 j g i,..., g k ) (g 1,..., g i,..., g k ) (g 1,..., g i g ±1 j,..., g k ) Walk randomly over this graph and select random g i.

18 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups Very good practical performance Biased in the case X = (Alt n ) n!/8 (Babai-Pak), although this bias does not affect the order statistics. In general, Γ k (G) is not connected Γ k (G) is connected for large k > 2 log 2 G. In that case (Babai) the diameter < C log 2 G

19 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups Conjecture. If G is simple, Γ k (G) is connected for k 3. Pak: If G n are simple, G n, then Γ k (G) has a connected component Γ k (G), Γ k (G) Γ k (G) 1 Pak: If k is sufficiently big, mixing time of a lazy random walk on Γ k (G) is bounded by a polynomial in k and log G.

20 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups Lubotzky - Pak: If Aut F k satisfies Kazhdan property (T), then mixing time of a random lazy walk on a component of Γ k (G) mix C(k) log 2 G. Conjecture. For k 4, Aut F k has (T).

21 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups Kazhdan property (T) G a topological group, Q G a compact set K = inf ρ inf max v 0 q Q ρ(q)(v) v v > 0 ρ: all unitary representations without fixed nonzero vectors Margulis: If Aut F k has (T) then Γ k (G) is an ε-expander, ε > K2 2 Q = ε(k) (Aut F k discrete, Q finite).

22 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups Expander: A graph Γ is an ε-expander if, for every B Γ, #B < 1 2 #Γ, # Lazy walk: { vertices connected to B, but not in B } ε #B Mixing time: number t of steps s.t. after t steps P (get at v) #Γ < 1 e v Γ

23 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups Andrews Curtis graph N G k (G, N) = { (h 1,..., h k ) h G 1,..., h G k = N} Edges: (x 1,..., x k ) (x 1,..., x i x j,..., x k ), i j (x 1,..., x k ) (x 1,..., x 1 i,..., x k ) (x 1,..., x k ) (x 1,..., x w i,..., x k), w G Conjecture. A random walk on k (G, N) provides a fast black box for N.

24 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups Myasnikov: k (F (m) n, F (m) ) is connected, k n n F (m) n free solvable group of class m The Andrews-Curtis Conjecture (1965): k (F k, F k ) is connected Some potential counterexamples (originating in topology) are killed by application of genetic algorithms, say, in F = x, y, (x 2 y 3, xyxy 1 x 1 y 1 ) (x, y) (example by Akbulut and Kirbi, 1985).

25 Alexandre Borovik Probabilistic and Non-Deterministic Computations in Finite Groups Current line of attack: study of k (G, G) for finite groups G. Theorem If G is a finite simple group then d( k (G, G)) < c k log G