a group theoretic approach to a family of 2 local finite groups constructed by levi and oliver we extend the notion of a p local finite group defined in blo03 to the notion of a p local group we define morphisms of p local groups obtaining thereby a category and we introduce the notion of a representation of a p local group via signalizer functors associated with groups we construct a chain g g 0 g 1 of 2 local finite groups via a representation of a chain g g 0 g 1 of groups such that g 0 is the 2 local finite group of the third conway sporadic group co 3 and for n 0 g n is one of the 2 local finite groups constructed by levi and oliver in lo02 we show that the direct limit g of g exists in the category of 2 local groups and that it is the 2 local group of the union of the chain g the 2 completed classifying space of g is shown to be the classifying space b di 4 of the exotic 2 compact group of dwyer and wilkerson dw93

fusion systems over a sylow p subgroup of text g _2 p g 2 p for s a sylow p subgroup of the group text g 2 p for p odd up to isomorphism of fusion systems we determine all saturated fusion systems mathcal f on s with o p mathcal f 1 for p ne 7 all such fusion systems are realized by finite groups whereas for p 7 there are 29 saturated fusion systems of which 27 are exotic

subcentric linking systems for part of this research the author was supported by the danish national research foundation through the centre for symmetry and deformation dnrf92

scaling solutions of smoluchowski s coagulation equation we investigate the structure of scaling solutions of smoluchowski s coagulation equation of the formc k t s t k s t wherec k t is the concentration of clusters of sizek at timet s t is the average cluster size and x is a scaling function for the rate constantk i j in smoluchowski s equation we make the very general assumption thatk i j is a homogeneous function of the cluster sizesi andj k i j a k ai aj for alla 0 but we restrict ourselves to kernels satisfyingk i j j 0 asj we show that gelation occurs if 1 and does not occur if 1 for all gelling and nongelling models we calculate the time dependence ofs t and we derive an equation for x we present a detailed analysis of the behavior of x at large and small values ofx for all models we find exponential large x behavior x ax e x asx and for different kernelsk i j algebraic or exponential small x behavior x bx or x exp cx asx 0

kinetics of gelation and universality the exact solution size distribution ck t and moments m t of smoluchowski s coagulation equation s model and of a modified equation f model with a coagulation rate k ij for i and j clusters is obtained for arbitrary ck 0 in the sol t phases where tc is the gel point the behaviour of ck t and m t is given for k co t co and t i the critical exponents critical amplitudes and scaling function that characterise the singularities near the non equilibrium phase transition are calculated for short range ck 0 i e all m a the f model belongs to the universality class of classical gelation theories and of bond percolation on cayley trees the s model does not

saturated fusion systems over 2 groups we develop methods for listing for a given 2 group s all nonconstrained centerfree saturated fusion systems over s these are the saturated fusion systems which could potentially include minimal examples of exotic fusion systems fusion systems not arising from any finite group to test our methods we carry out this program over four concrete examples two of order 2 7 and two of order 2 10 our long term goal is to make a wider more systematic search for exotic fusion systems over 2 groups of small order

computing via material topology optimisation we construct logical gates via topology optimisation aimed to solve a station problem of heat conduction of a conductive material layout values of logical variables are represented by high and low values of a temperature at given sites logical functions are implemented via the formation of an optimum layout of conductive material between the sites with loading conditions we implement and and xor gates and a one bit binary half adder

analogues of goldschmidt s thesis for fusion systems we extend the results of david goldschmidt s thesis concerning fusion in nite groups to saturated fusion systems t 2 syl 2 g then the exponent of z t and hence of t is bounded by a function of the nilpotence class of t he also includes in the write up a fusion factorization result for an arbitrary nite group involving 0 1 z and the thompson subgroup in this paper we generalize these results to arbitrary saturated fusion systems throughout this paper unless otherwise in dicated p denotes an arbitrary prime number n a nonnegative integer and p a nontrivial nite p group theorem 1 suppose p is of nilpotence class at most n p 1 1 andf is a saturated fusion system on p with op f 1 then z p has exponent at most p n

a characteristic subgroup for fusion systems abstract as a counterpart for the prime 2 to glauberman s zj theorem stellmacher proves that any nontrivial 2 group s has a nontrivial characteristic subgroup w s with the following property for any finite 4 free group g with s a sylow 2 subgroup of g and with o 2 g self centralizing the subgroup w s is normal in g we generalize stellmacher s result to fusion systems a similar construction of w s can be done for odd primes and gives rise to a glauberman functor

minimal fusion systems with a unique maximal parabolic abstract we define minimal fusion systems in a way that every non solvable fusion system has a section which is minimal minimal fusion systems can also be seen as analogs of thompson s n groups in this paper we consider a minimal fusion system f on a finite p group s that has a unique maximal p local subsystem containing n f s for an arbitrary prime p we determine the structure of a certain explicitly described p local subsystem of f if p 2 this leads to a complete classification of the fusion system f

saturated fusion systems with parabolic families abstract let g be a group a finite p subgroup s of g is a sylow p subgroup if every finite p subgroup of g is conjugate to a subgroup of s in this paper we examine the relations between the fusion system over s which is given by conjugation in g and a certain chamber system c on which g acts chamber transitively with chamber stabilizer n g s next we introduce the notion of a fusion system with a parabolic family and we show that a chamber system can be associated to such a fusion system we determine some conditions the chamber system has to fulfill in order to assure the saturation of the underlying fusion system we give an application to fusion systems with parabolic families of classical type

fusion systems over non resistant a2 groups abstract in this paper we study the saturated fusion systems f over potential non resistant a 2 groups of order greater than p 4 for each case in the classification f is constrained and realized by a finite group

generation of fusion systems of characteristic 2 type we prove that if is a saturated fusion system on a finite 2 group s then either is known or is generated by the normalizers of two canonically defined characteristic subgroups of s there are various corollaries for finite groups of characteristic 2 type

products in fusion systems abstract we revisit the notion of a product of a normal subsystem with a p subgroup as defined by aschbacher 2011 asc11 chapter 8 in particular we give a previously unknown more transparent construction

subgroup families controlling p local finite groups a p local finite group consists of a finite p group s together with a pair of categories which encode conjugacy relations among subgroups of s and which are modelled on the fusion in a sylow p subgroup of a finite group it contains enough information to define a classifying space which has many of the same properties as p completed classifying spaces of finite groups in this paper we examine which subgroups control this structure more precisely we prove that the question of whether an abstract fusion system mathcal f over a finite p group s is saturated can be determined by just looking at smaller classes of subgroups of s we also prove that the homotopy type of the classifying space of a given p local finite group is independent of the family of subgroups used to define it in the sense that it remains unchanged when that family ranges from the set of mathcal f centric mathcal f radical subgroups at a minimum to the set of mathcal f quasicentric subgroups at a maximum finally we look at constrained fusion systems analogous to p constrained finite groups and prove that they in fact all arise from groups

inverse problems of aggregation processes the coagulation frequency is the key ingredient in the population balance smoluchowski equation of coagulation kinetics an inverse problem is formulated to extract the coagulation frequency from transient size distributions when these distributions are self similar two numerical examples illustrate the procedure the first demonstrates the inverse problem for the recovery of singular coagulation frequencies while the second shows the procedure when self similarity is approximate transient droplet coagulation experiments in a turbulent flow field have been performed the resulting size distributions are observed to be self similar the inverse problem is used to determine the drop coagulation frequency this frequency shows significant deviation from the coagulation frequencies derived from simple models of drop drop interactions in a turbulent flow field

control of transfer and weak closure in fusion systems we show that k1 and k 1 control transfer in every fusion system on a finite p group when p 5 and that they control weak closure of elements in every fusion system on a finite p group when p 3 this generalizes results of g glauberman concerning finite groups

homotopy classification of self maps of bg via g actions to complete the classification of self maps of bg theorem 4 2 it remains to prove theorem 4 1 i e to show that the higher limits of certain functors n 1h 2 f p g z p wall vanish and that h0 mp g f i has the expected value in this section these limits h mp g hi will be calculated in the easy cases when g is a simply connected classical group i e one of the groups su n sp n or spin n or when i 2 the computation of h mp g hi for the simply connected exceptional lie groups will be put off until section 6 we begin with some general properties of higher limits let 6 be any small discrete category it is convenient when dealing with higher limits of functors on to let 4 moldenote the category of contravariant functors from v to ad then for any f e w muok we can identify

wreath products and representations of p local finite groups abstract given two finite p local finite groups and a fusion preserving morphism between their sylow subgroups we study the question of extending it to a continuous map between their classifying spaces the results depend on the construction of the wreath product of p local finite groups which is also used to study p local permutation representations

saturated fusion systems over mathcal a _2 groups in this paper we study the saturated fusion systems mathcal f over mathcal a 2 groups s of order greater than p 4 this study requires the analysis of the possible saturated fusion systems in terms of the mathcal f automorphisms of the possible mathcal f centric and mathcal f radical subgroups also for each case in the classification either s is resistant or mathcal f is constrained the classification identifies a large family of resistant groups complementing the results on resistant groups found by various authors such as green minh stancu and diaz ruiz viruel moreover the classification provides some interesting examples of saturated fusion systems which have an mathcal f essential abelian subgroup of index p

amalgams blocks weights fusion systems and finite simple groups abstract we associate an iterated amalgam of finite groups to a certain class of fusion systems on finite p groups p a prime in such a way that the p group of the fusion system is a maximal finite p subgroup of the resulting group unique up to conjugacy and furthermore the conjugation action of the resulting usually infinite group on p subgroups induces the original fusion system on the p group 1 in view of earlier work of puig and of broto castellana grodal levi and oliver c broto n castellana j grodal r levi r oliver subgroup families controlling p local finite groups proc london math soc 3 91 2005 325 354 the fusion systems we deal with include all saturated fusion systems the resulting amalgam has free normal subgroups of finite index and we examine the images of the group by its maximal free normal subgroups of finite index these images all contain isomorphic copies of the original p group and are generated by the images of the finite groups used in the amalgamation if there is no non trivial normal p subgroup of the fusion system equivalently if the iterated amalgam constructed has no non trivial normal p subgroup then the generalised fitting subgroup of each of these homomorphic images is a direct product of non abelian simple groups each of order divisible by p and if there is no proper non trivial strongly closed p subgroup in the fusion system then the generalised fitting subgroup of any of these finite groups is characteristically simple we note that the finite dimensional representation theory of this amalgam is almost by construction p locally determined in the case of the fusion system associated to a p block of a finite group we suggest strong links between block theoretic invariants of the above finite epimorphic images of the associated amalgam and of the original block we believe that the results of this paper offer the prospect of at least the beginnings of a structural explanation for some of the current numerical conjectures in block theory e g j l alperin weights for finite groups in proc sympos pure math vol 47 amer math soc providence 1987 pp 369 379 e c dade counting characters in blocks i invent math 109 1 1992 187 210

extensions of linking systems with p group kernel we study extensions of p local finite groups where the kernel is a p group in particular we construct examples of saturated fusion systems mathcal f which do not come from finite groups but which have normal p subgroups a vartriangleleft mathcal f such that mathcal f a is the fusion system of a finite group one of the tools used to do this is the concept of a transporter system which is modelled on the transporter category of a finite group and is more general than a linking system

the gluing problem in the fusion systems of the symmetric alternating and linear groups a solution to linckelmann s gluing problem in all fusion systems of blocks yields an alternative formulation of alperin s weight conjecture using bredon equivariant cohomology techniques we solve the gluing problem in fusion systems of blocks which are isomorphic to the fusion systems of the symmetric groups or the alternating groups or gld q or sld q at the defining characteristic

minami webb splittings and some exotic p local finite groups we give a new proof of the minami webb formula for classifying spaces of finite groups by exploiting symonds s resolution of webb s conjecture the methods are applicable to obtain a stable decomposition of minami s type for the classifying spaces of the three exotic p local finite groups which were introduced by ruiz and viruel at the prime 7 we obtain a similar decomposition for the classifying spaces of a family of exotic p local finite groups which were constructed by broto levi and oliver

kinetics of growing and coalescing droplets a class of mean field models for the patterns formed by growing and coalescing droplets on a line is introduced and solved all the models are found to have a scale invariant regime where the patterns are self similar in time

homotopy equivalences of p completed classifying spaces of finite groups we study homotopy equivalences of p completions of classifying spaces of finite groups to each finite group g and each prime p we associate a finite category p c g with the following properties two p completed classifying spaces bg p and bg p have the same homotopy type if and only if the associated categories p c g and p c g are equivalent furthermore the topological monoid aut bg p of self equivalences is determined by the self equivalences of the associated category p c g

the homotopy theory of fusion systems the main goal of this paper is to identify and study a certain class of spaces which in many ways behave like p completed classifying spaces of finite groups these spaces occur as the classifying spaces of certain algebraic objects which we call p local finite groups a p local finite group consists roughly speaking of a finite p group s and fusion data on subgroups of s encoded in a way explained below our starting point is our earlier paper blo on p completed classifying spaces of finite groups together with the axiomatic treatment by lluis puig pu pu2 of systems of fusion among subgroups of a given p group the p completion of a space x is a space xp which isolates the properties of x at the prime p and more precisely the properties which determine its mod p cohomology for example a map of spaces x f y induces a homotopy equivalence xp y p if and only if f induces an isomorphism in mod p cohomology and h xp fp h x fp in favorable cases if x is p good when g is a finite group the p completion bgp of its classifying space encodes many of the properties of g at p for example not only the mod p cohomology of bg but also the sylow p subgroup of g together with all fusion among its subgroups are determined up to isomorphism by the homotopy type of bgp our goal here is to give a direct link between p local structures and homotopy types which arise from them this theory tries to make explicit the essence of what it means to be the p completed classifying space of a finite group and at the same time yields new spaces which are not of this type but which still enjoy most of the properties a space of the form bgp would have we hope that the ideas presented here will have further applications and generalizations in algebraic topology but this theory also fits well with certain aspects of modular representation theory in particular it may give a way of constructing classifying spaces for blocks in the group ring of a finite group over an algebraically closed field of characteristic p a saturated fusion system f over a p group s consists of a set homf p q of monomorphisms for each pair of subgroups p q s which form a category under composition include all monomorphisms induced by conjugation in s and satisfy certain other axioms formulated by puig definitions 1 1 and 1 2 below in particular these axioms are satisfied by the conjugacy homomorphisms in a finite group we refer to pu and pu2 for more details of puig s work on saturated

the classification of p local finite groups over the extraspecial group of order p 3 and exponent p in this paper we classify the p local finite groups over p1 2 the extraspecial group of order p3 and exponent p for odd p this study reduces to the classification of the saturated fusion systems over p1 2 which will be characterized by the outer automorphism group the number of open image in new window radical subgroups and the automorphism group of each nontrivial open image in new window radical subgroup as part of this classification we obtain three new exotic 7 local finite groups