recognising simplicity of black box groups by constructing involutions and their centralisers abstract we investigate the complexity of constructing involutions and their centralisers in groups of lie type over finite fields of odd order and discuss applications to the problem of deciding whether a matrix group or a black box group of known characteristic is simple we show that if the characteristic is odd then simplicity can be recognised in monte carlo polynomial time

prime power graphs for groups of lie type abstract we associate a weighted graph g to each finite simple group g of lie type we show that with an explicit list of exceptions g determines g up to isomorphism and for these exceptions g nevertheless determines the characteristic of g this result was motivated by algorithmic considerations we prove that for any finite simple group g of lie type input as a black box group with an oracle to compute the orders of group elements g and the characteristic of g can be computed by a monte carlo algorithm in time polynomial in the input length the characteristic is needed as part of the input in a previous constructive recognition algorithm for g

gaussian bounds for degenerate parabolic equations abstract let a be a real symmetric degenerate elliptic matrix whose degeneracy is controlled by a weight w in the a 2 or qc class we show that there is a heat kernel w t x y associated to the parabolic equation w u t div a u and w t satisfies classic gaussian bounds w t x y c 1 t n 2 exp c 2 x y 2 t we then use this bound to derive a number of other properties of the kernel

cellular automata between sofic tree shifts we study the sofic tree shifts of a s where s is a regular rooted tree of finite rank in particular we give their characterization in terms of unrestricted rabin automata we show that if x a s is a sofic tree shift then the configurations in x whose orbit under the shift action is finite are dense in x and as a consequence of this we deduce that every injective cellular automata t x x is surjective moreover a characterization of sofic tree shifts in terms of general rabin automata is given we present an algorithm for establishing whether two unrestricted rabin automata accept the same sofic tree shift or not this allows us to prove the decidability of the surjectivity problem for cellular automata between sofic tree shifts we also prove the decidability of the injectivity problem for cellular automata defined on a tree shift of finite type

large element orders and the characteristic of lie type simple groups we show that the three largest element orders determine the characteristic of lie type simple groups of odd characteristic this result was motivated by algorithmic needs in computations with matrix groups

on classifying all full factorisations and multiple factorisations of the finite almost simple groups abstract we present some general results on factorisations of almost simple groups these results are consequences of the classification of all maximal factorisations of almost simple groups due to liebeck saxl and the second author and are needed for applications to the theory of quasiprimitive permutation groups

tensor products are projective geometries abstract given a finite dimensional vector space v we construct a family of projective geometries whose flats are certain subspaces of v and show that there is a one to one correspondence between this family of projective geometries and the set of equivalence classes of tensor decompositions of v this provides a practical method for finding a tensor decomposition of a finite dimensional kg module or proving that no nontrivial tensor decomposition exists

short presentations for three dimensional unitary groups abstract we give a presentation of length o log2 g for the groups g psu3 q this result has applications in recent algorithms to compute the structure of permutation groups and matrix groups

generation of finite classical groups by pairs of elements with large fixed point spaces abstract we study good elements in finite 2 n dimensional classical groups g namely t is a good element if o t is divisible by a primitive prime divisor of q n 1 for the relevant field order q and t fixes pointwise an n space the group sl 2 n q contains such elements and they are present in su 2 n q sp 2 n q so 2 n e q only if n is odd even even respectively we prove that there is an absolute positive constant c such that two random conjugates of t generate g with probability at least c if g so 2 n e 2 and g sp 2 n q with q even in the exceptional case g sp 2 n q with q even two conjugates of t never generate g in this case we prove that two random conjugates of t generate a subgroup so 2 n e q with probability at least c the results underpin analysis of new constructive recognition algorithms for classical groups in even characteristic which succeed where methods utilising involution centralisers are not available

weighted sobolev poincare inequalities and pointwise estimates for a class of degenerate elliptic equations in this paper we prove a sobolev poincare inequality for a class of function spaces associated with some degenerate elliptic equations these estimates provide us with the basic tool to prove an invariant harnack inequality for weak positive solutions in addition holder regularity of the weak solutions follows in a standard way let y z 7 ai aij9 a be a second order degenerate elliptic operator in divergence form with measurable coefficients in this paper we shall obtain pointwise estimates for the weak solutions of su 0 h61der continuity of the weak solutions and harnack inequality for nonnegative solutions let us recall that the original results for elliptic operators were obtained by de giorgi nash and moser an extensive bibliography about the degenerate case can be found in fli fl2 fs to introduce the results of the present paper let us recall some recent results in fl1 fl2 a suitable metric d is associated with the differential operator y in such a way that we obtain a new geometry which is natural for the degenerate operator as the euclidean geometry is natural for the laplace operator or more precisely as a suitable riemannian geometry is natural for a secondorder elliptic operator in the smooth case this idea is contained in many papers we refer to fp nsw the basic results in fl 1 fl2 are obtained via a precise description of this geometry under suitable technical hypotheses on the coefficients whose aim is to give a nonsmooth formulation of the hormander hypoellipticity condition for sum of squares operators we note that the same idea is used in nsw s j v to obtain pointwise estimates for sum of squares operators on the other hand a different class of degenerate elliptic operators is considered in fks instead of a geometrical degeneracy a measure degeneracy is allowed a typical example of this class is given by yu div wo x vu where cl is a weight function belonging to the a2 class of muckenhoupt unified results for a class containing both the operators in fl 1 and in fks have received by the editors august 1 1989 1980 mathematics subject classification 1985 revision primary 46e35 35j70 partially supported by g n a f a of c n r and m u r s t italy 1991 american mathematical society 0002 9947 91 1 00 25 per page

the kato problem for operators with weighted ellipticity we consider second order operators lw w 1 divawr with ellipticity controlled by a muckemphout a2 weight w we prove that the kato square root estimate l 1 2 w f l2 w kr fkl2 w holds in the weighted space l 2 w

recognising the suzuki groups in their natural representations abstract under the assumption of a certain conjecture for which there exists strong experimental evidence we produce an efficient algorithm for constructive membership testing in the suzuki groups sz q where q 2 2 m 1 for some m 0 in their natural representations of degree 4 it is a las vegas algorithm with running time o log q field operations and a preprocessing step with running time o log q log log q field operations the latter step needs an oracle for the discrete logarithm problem in f q we also produce a recognition algorithm for sz q x this is a las vegas algorithm with running time o x 2 field operations finally we give a las vegas algorithm that given x h sz q for some h gl 4 q finds some g such that x g sz q the running time is o log q log log q x field operations implementations of the algorithms are available for the computer system magma

involution centralisers in finite unitary groups of odd characteristic abstract we analyse the complexity of constructing involution centralisers in unitary groups over fields of odd order in particular we prove logarithmic bounds on the number of random elements required to generate a subgroup of the centraliser of a strong involution that contains the last term of its derived series we use this to strengthen previous bounds on the complexity of recognition algorithms for unitary groups in odd characteristic our approach generalises and extends two previous papers by the second author and collaborators on strong involutions and regular semisimple elements of linear groups

the solution of the kato problem for degenerate elliptic operators with gaussian bounds we prove the kato conjecture for degenerate elliptic operators on r n more precisely we consider the divergence form operator lw w 1 diva where w is a muckenhoupt a2 weight and a is a complex valued n n matrix such that w 1 a is bounded and uniformly elliptic we show that if the heat kernel of the associated semigroup e tlw satisfies gaussian bounds then the weighted kato square root estimate kl 1 2 w f k l2 w k f k l2 w holds

strong involutions in finite special linear groups of odd characteristic abstract let t be an involution in gl n q whose fixed point space e has dimension k between n 3 and 2 n 3 for each g gl n q such that t t g has even order t t g contains a unique involution z g which commutes with t we prove that with probability at least c log n for some c 0 the restriction z g e is an involution on e with fixed point space of dimension between k 3 and 2 k 3 this result has implications in the analysis of the complexity of recognition algorithms for finite classical groups in odd characteristic we discuss how similar results for involutions in other finite classical groups would solve a major open problem in our understanding of the complexity of constructing involution centralisers in those groups

the structure of rank 3 permutation modules for o2n 2 and um 2 acting on nonsingular points abstract we study the odd characteristic structure of permutation modules for the rank 3 natural actions of o 2 n 2 n 3 and u m 2 m 4 on nonsingular points of their standard modules

the solution of the kato problem for divergence form elliptic operators with gaussian heat kernel bounds which in turnhas applications to the perturbation theory for certain classes of hyperbolicequations see 16 we remark that 1 3 is equivalent to the oppositeinequality for the square root of the adjoint operator l in 13 14 kato conjectured that an abstract version of 1 3 might hold for regularly accretive operators see 14 18 for the details a counter example to this abstract conjecture was obtained by mcintosh 17 whothen reformulated the conjecture in the following form bearing in mind thatkato s interest in the problem had been motivated by the special case ofelliptic di erential operators conjecture 1 4 the estimate 1 3 holds for lde ned as in 1 2 for anyl

classifying circular cellular automata we introduce a hierarchy of linear cellular automata based on their limiting behavior on spatially periodic configurations we show that it is undecidable to which class in the hierarchy a cellular automaton belongs in particular it is undecidable whether all spatially periodic configurations evolve to a fixed point furthermore there is no computable bound on the period lengths of these configurations our arguments are based on a non standard simulation of turing machines on circular cellular automata

writing projective representations over subfields let g x be an absolutely irreducible subgroup of gl d k and let f be a proper subfield of the finite field k we present a practical algorithm to decide constructively whether or not g is conjugate to a subgroup of gl d f k where k denotes the centre of gl d k if the derived group of g also acts absolutely irreducibly then the algorithm is las vegas and costs o x d3 d2log f arithmetic operations in k this work forms part of a recognition project based on aschbacher s classification of maximal subgroups of gl d k

finding involutions in finite lie type groups of odd characteristic abstract let g be a finite group of lie type in odd characteristic defined over a field with q elements we prove that there is an absolute and explicit constant c such that if g is a classical matrix group of dimension n 2 then at least c log n of its elements are such that some power is an involution with fixed point subspace of dimension in the interval n 3 2 n 3 if g is exceptional or g is classical of small dimension then for each conjugacy class c of involutions we find a very good lower bound for the proportion of elements of g for which some power lies in c

nearly optimal algorithms for canonical matrix forms a las vegas type probabilistic algorithm is presented for finding the frobenius canonical form of an n times n matrix t over any field kk the algorithm requires softo mm n mm n cdot log n o 1 operations in kk where o mm n operations in kk are sufficient to multiply two n times n matrices over kk this nearly matches the lower bound of omega mm n operations in kk for this problem and improves on the o n 4 operations in kk required by the previously best known algorithms a fast parallel implementation of the algorithm is also demonstrated for the frobenius form which is processor efficient on a pram as an application we give an algorithm to evaluate a polynomial g in kk x at t which requires only softo mm n operations in kk when deg g leq n 2 other applications include sequential and parallel algorithms for computing the minimal and characteristic polynomials of a matrix the rational jordan form of a matrix for testing whether two matrices are similar and for matrix powering which are substantially faster than those previously known

constructive recognition of sl3 q abstract we present a constructive recognition algorithm for groups of lie type sl 3 q this is a necessary component for constructive recognition algorithms of quasisimple groups of lie type

constructive recognition of classical groups in even characteristic abstract let g x gl d f be a classical group in its natural representation defined over a finite field f of even characteristic we present las vegas algorithms to construct standard generators for g which permit us to write an element of g as a straight line program in x and to construct an involution as a straight line program in x if f 4 then the algorithms run in time polynomial in the size of the input subject to the existence of a discrete logarithm oracle for f

the trivial intersection problem for characters of principal indecomposable modules abstract for a prime p and a finite group g let p g denote the complex character associated to the projective indecomposable module in characteristic p with trivial head let irr p g denote the set of irreducible characters occurring as constituents in p g we characterize all finite simple groups which satisfy irr p g irr q g 1 g for all primes p q

adjoint representations of black box groups psl2 fq abstract given a black box group y encrypting psl 2 f over an unknown field f of unknown odd characteristic p and a global exponent e for y that is an integer e such that y e 1 for all y y we present a las vegas algorithm which constructs a unipotent element in y the running time of our algorithm is polynomial in log e this answers the question posed by babai and beals in 1999 we also find the characteristic of the underlying field in time polynomial in log e and linear in p all our algorithms are randomized furthermore we construct in time polynomial in log e a black box group x encrypting pgl 2 f so 3 f over the same field as y its subgroup y of index 2 isomorphic to y and a polynomial in log e time isomorphism y y a black box field k and polynomial time in log e isomorphisms so 3 k x so 3 k if in addition we know p and the standard explicitly given finite field f q isomorphic to the field on which y is defined then we construct in time polynomial in log e isomorphism so 3 f q so 3 k unlike many papers on black box groups our algorithms make no use of additional oracles other than the black box group operations moreover our result acts as an sl 2 oracle in the black box group theory we implemented our algorithms in gap and tested them for groups such as psl 2 f for f 115756986668303657898962467957 a prime number

polynomial growth estimates for multilinear singular integral operators on demontre un theoreme pour des mesures de carleson et on l applique a l operateur de kato en dimension 1 on demontre un critere de limitation general on donne une application aux commutateurs de coolderon on etudie les d commutateurs

fast constructive recognition of black box orthogonal groups abstract we present an algorithm that constructively recognises when a given black box group is a nontrivial homomorphic image of the orthogonal group e d q for known e d and q the algorithm runs in polynomial time assuming oracles for handling sl 2 q subgroups and discrete logarithms in f q

uniform 2 k generation of the 4 dimensional classical groups abstract in this paper we study the 2 k generation of the finite classical groups sl 4 q sp 4 q su 4 q 2 and their projective images here k is the order of an arbitrary element of sl 2 q subject to the necessary condition k 3 when q is even we allow also k 4

binary codes derived from the hoffman singleton and higman sims graphs some binary linear codes of length 50 and 100 are constructed using the adjacency matrices of the hoffman singleton graph and the higman sims graph some of the codes are optimal or nearly optimal for the given length and dimension the dual codes admit majority logic decoding

a language hierarchy and kitchens type theorem for self similar groups abstract we generalize the notion of self similar groups of infinite tree automorphisms to allow for groups which are defined on a tree but may not act faithfully on it the elements of such a group correspond to labeled trees which may be recognized by a tree automaton e g rabin buchi etc or considered as elements of a tree shift e g of finite type sofic as in symbolic dynamics we give examples to show how self similar groups defined in this way can be separated into different tree language hierarchies as the main result extending the classical result of kitchens on one dimensional group shifts we provide a sufficient condition for a self similar group whose elements form a sofic tree shift to be a tree shift of finite type as an application we show that the closures of certain self similar groups of rooted k ary tree automorphisms that satisfy an algebraic law are not rabin recognizable that is they can not be described within the second order theory of k successors in both the main result and in the application a crucial role is played by a distinguished branched subgroup structure of the groups under consideration

on products of involutions in finite groups of lie type in even characteristic abstract let g be a finite simple group of lie type in characteristic 2 and t g an involution we provide a lower bound for the proportion of elements g g such that t t g has odd order this has applications to the theory of recognition algorithms for such groups

the solution of the kato square root problem for second order elliptic operators on rn we prove the kato conjecture for elliptic operators on jfin more precisely we establish that the domain of the square root of a uniformly complex elliptic operator l div av with bounded measurable coefficients in ietn is the

cellular automata and strongly irreducible shifts of finite type if a is a finite alphabet and is a finitely generated amenable group ceccherini silberstein machi and scarabotti have proved that a local transition function defined on the full shift a is surjective if and only if it is pre injective this equivalence is the so called garden of eden theorem on the other hand when is the group of the integers the theorem holds in the case of irreducible shifts of finite type as a consequence of a theorem of lind and marcus but it no longer holds in the two dimensional case recently gromov has proved a goe like theorem in the much more general framework of the spaces of bounded propagation in this paper we apply gromov s theorem to our class of spaces proving that all the properties required in the hypotheses of this theorem are satisfied we give a definition of strong irreducibility that together with the finite type condition it allows us to prove the goe theorem for the strongly irreducible shifts of finite type in a provided that is amenable finally we prove that the bounded propagation property for a shift is strictly stronger than the union of strong irreducibility and finite type condition

constructing composition factors for a linear group in polynomial time abstract we present a las vegas polynomial time algorithm that takes as input a subgroup of gl d f q and subject to the existence of certain oracles determines its composition factors provided that none of those factors is isomorphic to one of b 2 2 2 2 k 1 f 4 2 2 2 k 1 d 4 3 2 k or g 2 2 3 2 k 1 for any k

a black box group algorithm for recognizing finite symmetric and alternating groups i we present a las vegas algorithm which for a given black box group known to be isomorphic to a symmetric or alternating group produces an explicit isomorphism with the standard permutation representation of the group this algorithm has applications in computations with matrix groups and permutation groups in this paper we handle the case when the degree n of the standard permutation representation is part of the input in a sequel we shall treat the case when the value of n is not known in advance as an important ingredient in the theoretical basis for the algorithm we prove the following result about the orders of elements of s n the conditional probability that a random element s n is an n cycle given that n 1 is at least 1 10

black box groups isomorphic to pgl 2 2e abstract a deterministic polynomial time algorithm constructs an isomorphism between pgl 2 2 e and a black box group to which it is isomorphic

fast constructive recognition of black box symplectic groups abstract we present an algorithm which for any given black box group g isomorphic to sp d q constructs explicit inverse isomorphisms between g and the standard copy of sp d q we also report on the performance of an implementation of this algorithm in the group theory system gap

the nilpotency problem of one dimensional cellular automata the limit set of a celullar automaton consists of all the configurations of the automaton that can appear after arbitrarily long computations it is known that the limit set is never empty it contains at least one homogeneous configuration a ca is called nilpotent if its limit set contains just one configuration the present work proves that it is algorithmically undecidable whether a given one dimensional cellular automaton is nilpotent the proof is based on a generalization of the well known result about the undecidability of the tiling problem of the plane the generalization states that the tiling problem remains undecidable even if one considers only so called nw deterministic tile sets that is tile sets in which the left and upper neighbors of each tile determine the tile uniquely the nilpotency problem is known to be undecidable for d dimensional ca for d geq 2 the result is the basis of the proof of rice s theorem for ca limit sets which states that every nontrivial property of limit set

rank 3 permutation modules of the finite classical groups abstract the cross characteristic permutation modules for the actions of the finite classical groups on singular 1 spaces of their natural modules are studied the composition factors and submodule lattices are determined

short presentations for finite groups we conjecture that every finite groupghas a short presentation in terms of generators and relations in the sense that the totallengthof the relations is log g o 1 we show that it suffices to prove this conjecture for simple groups motivated by applications in computational complexity theory we conjecture that for finite simple groups such a short presentation is computable in polynomial time from the standard name ofg assuming in the case of lie type simple groups overgf pm that an irreducible polynomialfof degreemovergf p and a primitive root ofgf pm are given we verify this stronger conjecture for all finite simple groups except for the three families of rank 1 twisted groups we do not handle the unitary groupspsu 3 q 2a2 q the suzuki groupssz q 2b2 q and the ree groupsr q 2g2 q in particular all finite groups g without composition factors of these types have presentations of length o log g 3 for groups of lie type normal or twisted of rank 2 we use a reduced version of the curtis steinberg tits presentation

recognizing the characteristic of a simple group of lie type from its probabilistic zeta function abstract we find a method to recognize the characteristic of a simple group of lie type g from its dirichlet polynomial p g s this is enough to complete the proof of the following statement if g is a simple group h is a finite group and p g s p h s then h frat h g

periodic simple groups of finitary linear transformations a group is locally finite if every finite subset generates a finite subgroup a group of linear transformations is finitary if each element minus the identity is an endomorphism of finite rank the classification and structure theory for locally finite simple groups splits naturally into two cases those groups that can be faithfully represented as groups of finitary linear transformations and those groups that are not finitary linear this paper completes the finitary case we classify up to isomorphism those infinite locally finite simple groups that are finitary linear but not linear

representations of the alternating group which are irreducible over subgroups ii nsf dms 1161094 dms 0901241 dms 1201374 humboldt foundation simons foundation 204181 simons foundation fellowship 305247

the theory of ends pushdown automata and second order logic abstract a class of edge labeled graphs called context free are defined according to their behavior at infinity such graphs are generalizations of cayley graphs of context free groups they are also shown to be definable in a very natural way in terms of push down automata using rabin s theorem on the monadic second order theory of the finite binary tree these graphs are also shown to have a decidable monadic second order theory questions about tiling systems and cellular automata operating on these graphs are decidable even when the analogous questions are not when the systems operate on a two dimensional grid

constructive recognition of mathrm psl 2 q existing black box and other algorithms for explicitly recognising groups of lie type over gf q have asymptotic running times which are polynomial in q whereas the input size involves only log q this has represented a serious obstruction to the efficient recognition of such groups recently brooksbank and kantor devised new explicit recognition algorithms for classical groups these run in time that is polynomial in the size of the input given an oracle that recognises psl 2 q explicitly the present paper in conjunction with an earlier paper by the first two authors provides such an oracle the earlier paper produced an algorithm for explicitly recognising sl 2 q in its natural representation in polynomial time given a discrete logarithm oracle for gf q the algorithm presented here takes as input a generating set for a subgroup g of gl d f that is isomorphic modulo scalars to psl 2 q where f is a finite field of the same characteristic as gf q it returns the natural representation of g modulo scalars since a faithful projective representation of psl 2 q in cross characteristic or a faithful permutation representation of this group is necessarily of size that is polynomial in q rather than in log q elementary algorithms will recognise psl 2 q explicitly in polynomial time in these cases given a discrete logarithm oracle for gf q our algorithm thus provides the required polynomial time oracle for recognising psl 2 q explicitly in the remaining case namely for representations in the natural characteristic this leads to a partial solution of a question posed by babai and shalev if g is a matrix group in characteristic p determine in polynomial time whether or not op g is trivial

decidability of irreducible tree shifts of finite type we reveal an algorithm for determining the complete prefix code irreducibility cpc irreducibility of dyadic trees labeled by a finite alphabet by introducing an extended directed graph representation of tree shift of finite type tsft we show that the cpc irreducibility of tsfts is related to the connectivity of its graph representation which is a similar result to one dimensional shifts of finite type

the automorphism groups of the generalized quadratic residue codes we give the full automorphism groups as groups of semiaffine transformations of the extended generalized quadratic residue codes we also present a proof of the gleason prange theorem for the extended generalized quadratic residue codes that relies only on their definition and elementary theory of linear characters

black box exceptional groups of lie type ii abstract if a black box group g is known to be isomorphic to an exceptional simple group g of lie type of twisted rank 1 other than any f 4 2 2 e or d 4 3 2 e over a field of known size q and if suitable sl 2 and discrete log oracles are available when q is odd then a polynomial time las vegas algorithm is given that produces a constructive isomorphism between g and g

a recognition algorithm for special linear groups contain th speciase l linea sl rfr grou q pamplifying the problem he said he would be happy to have a program that couldbe used fo r genera l purpose practical computatio in th range e rf n 30 charle ssims suggested tha onte should be more ambitiou ass to degree sa y d 70 bu trestrict onesel tfo the case q 2 in this pape wr e shall propos ae method whichwe believ teo be practica fol r value osf d up t somewhero e aroun 60d or 70 andwhich is no particularlt y sensitiv to thee size of q w muse t emphasize howeverthat our ideas hav noe t yet at th time e of writing been implemente andd tested until adequate evidenc ha accumulates e d such claim nost mus b takee t n veryseriously ou algorithr ims partly probabilistic it whes respons n ies positiv iettells us that the group generate bdy x certainly does contai sl rfn q but if theoutput is negative that assertion will usually b onle mady e with probabilit 1y e where e is a pre determined small positive number the analogous proble fomr permutatio n group is tso decide fo r a given se t xof permutation osf a set o sizf e n whethe thr e subgroup generate bdy x containsthe alternating group alt n an elegan t probabilistic algorith fo recognizinr m galternating and symmetric group hass been in use fo manr y years le t u cals l anelement of th symmetrie c group sym n if on primae of cycle its has prims elength p non oef the othe r cycles have length divisibl by p ane d ps n 3 themethod exploit thse following facts i if g i alt zs or sym n the an good proportio onf th elemente s o f g areprima ii if g does no contait n alt n and if g i primitivs a e permutation groupthen g contains no such elements the forme irs no hart d to prove see fo exampler 20 lectur e 11 th lattee rassertion comes from a well known old theorem of jordan see fo exampler 30 theorem 13 9 accordingl the ide ya i s to tes it f g i primitives whic cahn bedone pretty quickl byy fo exampler a method of atkinson i f no thet n certainlyg is neither alt n no sym n r i f g i primitives th e crucial ste isp to make asuitable numbe orf independent random selection of elements osf g no mant yare needed see fo r exampl 20 e som if e prima permutation show up thens gis identified correctl as alt ny o r sym n i f not the n with high probabilit g ydoes not contain alt n thus ther is a smale l probability tha the algoritht m willfail to identify alt n or sym n this procedur wa firsts e programme ford

recognising the small ree groups in their natural representations we present las vegas algorithms for constructive recognition and constructive membership testing of the ree groups g22 q ree q where q 32m 1 for some m 0 in their natural representations of degree 7 the input is a generating set x gl 7 q the constructive recognition algorithm is polynomial time given a discrete logarithm oracle the constructive membership testing consists of a pre processing step that only needs to be executed once for a given x and a main step the latter is polynomial time and the former is polynomial time given a discrete logarithm oracle implementations of the algorithms are available for the computer algebra system magma

recognition of finite exceptional groups of lie type let q be a prime power and let g be an absolutely irreducible subgroup of gld f where f is a finite field of the same characteristic as fq the field of q elements assume that g g q a quasisimple group of exceptional lie type over fq which is neither a suzuki nor a ree group we present a las vegas algorithm that constructs an isomorphism from g to the standard copy of g q if g 6 3d4 q with q even then the algorithm runs in polynomial time subject to the existence of a discrete log oracle

primitive groups graph endomorphisms and synchronization the third author has been partially supported by the fundacao para a ciencia e a tecnologia through the project cemat ciencias uid multi 04621 2013

black box exceptional groups of lie type if a black box group is known to be isomorphic to an exceptional simple group of lie type of twisted rank 1 other than any f4 q over a field of known size a las vegas algorithm is given to produce a constructive isomorphism in view of its timing this algorithm yields an upgrade of all known nearly linear time monte carlo permutation group algorithms to las vegas algorithms when the input group has no composition factor isomorphic to any group f4 q or g2 q

effective black box constructive recognition of classical groups abstract we describe a black box las vegas algorithm to construct standard generators for classical groups defined over finite fields we assume that the field has size at least 4 and that oracles to solve certain problems are available subject to these assumptions the algorithm runs in polynomial time a practical implementation of our algorithm is distributed with the computer algebra system magma