blocks of small defect abstract let g be a finite solvable group let p be a prime such that p 5 and o p g 1 and we denote g p p n then g contains a block of defect less than or equal to 3 n 5 let g be a finite solvable group and let p a be the largest power of p dividing 1 for an irreducible character of g we show that g f g p p 3 a for p 5

brauer characters and normal sylow p subgroups abstract in this paper we study some variations of the well known ito michler theorem for p brauer characters using various inequalities involving the p brauer character degrees of finite groups several new criteria for the existence of a normal sylow p subgroup of finite groups are obtained using the p parts and p parts of the p brauer character degrees

pseudo drazin inverses in associative rings and banach algebras abstract motivated by strongly regular elements and quasipolar elements we introduce the concept of pseudopolar elements an element a r is called pseudopolar if there exists p r such that p 2 p comm 2 a a p u r and a k p j r for some positive integer k this concept can be used exactly to define a pseudo drazin inverse in associative rings and banach algebras we connect pseudopolar rings with strongly regular rings semiregular rings uniquely strongly clean rings and uniquely bleached local rings some basic properties of pseudo drazin inverses are obtained in associative rings and banach algebras

characters of relative p degree over normal subgroups let z be a normal subgroup of a finite group g let irr z be an irreducible complex character of z and let p be a prime number if p does not divide the integers 1 1 for all irr g lying over then we prove that the sylow p subgroups of g z are abelian this theorem which generalizes the gluck wolf theorem to arbitrary finite groups is one of the principal obstacles to proving the celebrated brauer height zero conjecture

rings with internal cancellation in this paper we study the class of rings that satisfy internal direct sum cancellation with respect to their 1 sided ideals these are known to be precisely the rings in which regular elements are unitregular further characterizations for such ic rings are given in terms of suitable versions of stable range conditions and unique generator properties of idempotent generated right ideals this approach leads to a uniform treatment of many of the known characterizations for an exchange ring to have stable range 1 rings whose matrix rings are ic turn out to be precisely those rings whose finitely generated projective modules satisfy cancellation we also offer a couple of hidden characterizations of unit regular elements in rings that shed some new light on the relation between similarity and pseudo similarity in monoids as well as in rings the paper concludes with a treatment of ideals for which idempotents lift modulo all 1 sided subideals an appendix by r g swan 1 on the failure of cancellation for finitely generated projective modules over complex group algebras shows that such algebras are in general not ic

defect zero p blocks for finite simple groups we classify those finite simple groups whose brauer graph or decomposition matrix has a p block with defect 0 completing an investigation of many authors the only finite simple groups whose defect zero p blocks remained unclassified were the alternating groups an here we show that these all have a p block with defect 0 for every prime p 5 this follows from proving the same result for every symmetric group sn which in turn follows as a consequence of the t core partition conjecture that every non negative integer possesses at least one t core partition for any t 4 for t 17 we reduce this problem to lagrange s theorem that every non negative integer can be written as the sum of four squares the only case with t 17 that was not covered in previous work was the case t 13 this we prove with a very different argument by interpreting the generating function for t core partitions in terms of modular forms and then controlling the size of the coefficients using deligne s theorem nee the weil conjectures we also consider congruences for the number of p blocks of sn proving a conjecture of garvan that establishes certain multiplicative congruences when 5 p 23 by using a result of serre concerning the divisibility of coefficients of modular forms we show that for any given prime p and positive integer m the number of p blocks with defect 0 in sn is a multiple of m for almost all n we also establish that any given prime p divides the number of p modularly irreducible representations of sn for almost all n 1996 american mathematical society

boundary behavior and interior holder regularity of the solution to nonlinear stochastic partial differential equation driven by space time white noise abstract we present unique solvability result in weighted sobolev spaces of the equation u t a u x x b u x c u u 1 b t 0 x 0 1 given with initial data u 0 u 0 and zero boundary condition here 0 1 2 b is a space time white noise and the coefficients a b c and are random functions depending on t x we also obtain various interior holder regularities and boundary behaviors of the solution for instance if the initial data is in appropriate l p space then for any small e 0 and t almost surely 1 1 2 u c t x 1 4 2 e 1 2 e 0 t 0 1 1 2 where x is the distance from x to the boundary taking one gets the maximal holder exponents in time and space which are 1 4 2 e and 1 2 e respectively also letting 1 2 one gets better decay or behavior near the boundary

finite groups whose prime graphs are regular abstract let g be a finite group and let irr g be the set of all irreducible complex characters of g let cd g be the set of all character degrees of g and denote by g the set of primes which divide some character degrees of g the prime graph g associated to g is a graph whose vertex set is g and there is an edge between two distinct primes p and q if and only if the product pq divides some character degree of g in this paper we show that the prime graph g of a finite group g is 3 regular if and only if it is a complete graph with four vertices

finite group elements where no irreducible character vanishes in this paper we consider elements x of a finite group g with the property that x 0 for all irreducible characters of g if g is solvable and x has odd order we show that x must lie in the fitting subgroup f g

p parts of brauer character degrees abstract let g be a finite group and let p be an odd prime under certain conditions on the p parts of the degrees of its irreducible p brauer characters we prove the solvability of g as a consequence we answer a question proposed by b huppert in 1991 if g has exactly two distinct irreducible p brauer character degrees then is g solvable we also determine the structure of non solvable groups with exactly two irreducible 2 brauer character degrees

on the length of the conjugacy classes of finite groups abstract let g be a finite group the question of how certain arithmetical conditions on the degrees of the irreducible characters of g influence the group structure was studied by several authors our purpose here is to impose analogous conditions on the lengths of the conjugacy classes of g and to describe the group structure under these conditions

generalized inverses uniqueness proofs and three new classes abstract given any ring r with 1 and any a b c r then generalizing ideas of j j koliha and p patricio in 2002 and of z wang and j chen in 2012 a is called b c pseudo polar if there exists an idempotent p r such that 1 p b r j r c j p b and c p j where j denotes the jacobson radical of r and p lies in the second commutant of a this p is shown to be unique whenever it exists a new outer generalized inverse y of a called the b c pseudo inverse of a is also defined and the existence of y is shown to imply that a is b c pseudo polar and hence that y is itself unique generalizing results of koliha patricio wang and chen further connections between the b c pseudo polar and b c pseudo invertible properties are found and the b c pseudo invertibility of a 1 a 2 is shown to imply a corresponding property for a 2 a 1 two further types of uniquely defined outer generalized inverses are also introduced

bessel spdes and renormalised local times in this article we prove integration by parts formulae ibpfs for the laws of bessel bridges from 0 to 0 over the interval 0 1 of dimension smaller than 3 as an application we construct a weak version of a stochastic pde having the law of a one dimensional bessel bridge i e the law of a reflected brownian bridge as reversible measure the dimension 1 being particularly relevant in view of applications to scaling limits of dynamical critical pinning models we also exploit the ibpfs to conjecture the structure of the stochastic pdes associated with bessel bridges of all dimensions smaller than 3

an answer to a question of isaacs on character degree graphs abstract let n be a normal subgroup of a finite group g we consider the graph g n whose vertices are the prime divisors of the degrees of the irreducible characters of g whose kernel does not contain n and two vertices are joined by an edge if the product of the two primes divides the degree of some of the characters of g whose kernel does not contain n we prove that if g n is disconnected then g n is solvable this proves a strong form of a conjecture of isaacs

a distribution function valued spde and its applications abstract in this paper we further study the stochastic partial differential equation first proposed by xiong 22 under localized conditions on its coefficients we prove a comparison theorem on its solutions and show that the solution is in fact distribution function valued we also establish pathwise uniqueness of the solution as applications we obtain the well posedness of martingale problems for two classes of measure valued diffusions interacting super brownian motions and interacting fleming viot processes properties of the two superprocesses such as the existence of density fields and the survival extinction behaviors are also studied

a local conjecture on brauer character degrees of finite groups recently a new conjecture on the degrees of the irreducible brauer characters of a finite group was presented by w willems in this paper we propose a local version of this conjecture for blocks b of finite groups giving a lower bound for 1 2 where the sum runs through the set of irreducible brauer characters of b in terms of invariants of b a slight reformulation leads to interesting open questions about traces of cartan matrices of blocks we show that the local conjecture is true for blocks with one simple module blocks of p solvable groups and blocks with cyclic defect groups it also holds for many further examples of blocks of sporadic groups symmetric groups or groups of lie type finally we prove that the conjecture is true for blocks of tame representation type

on thompson s theorem let p be a prime divisor of the order of a finite group g thompson 1970 j algebra14 129 134 has proved the following remarkable result a finite group g is p nilpotent if the degrees of all its nonlinear irreducible characters are divisible by p in fact in that case g is solvable in this note we prove that a group g having only one nonlinear irreducible character of p degree is a cyclic extension of thompson s group this result is a consequence of the following theorem a nonabelian simple group possesses two nonlinear irreducible characters 1 and 2 of distinct degrees such that p does not divide 1 1 2 1 here p is arbitrary but fixed our proof depends on the classification of finite simple groups some properties of solvable groups possessing exactly two nonlinear irreducible characters of p degree are proved some open questions are posed

on the orders of zeros of irreducible characters abstract let g be a finite group and p a prime number we say that an element g in g is a vanishing element of g if there exists an irreducible character of g such that g 0 the main result of this paper shows that if g does not have any vanishing element of p power order then g has a normal sylow p subgroup also we prove that this result is a generalization of some classical theorems in character theory of finite groups

groups whose prime graphs have no triangles abstract let g be a finite group and let cd g be the set of all complex irreducible character degrees of g let g be the set of all primes which divide some character degree of g the prime graph g attached to g is a graph whose vertex set is g and there is an edge between two distinct primes u and v if and only if the product uv divides some character degree of g in this paper we show that if g is a finite group whose prime graph g has no triangles then g has at most 5 vertices we also obtain a classification of all finite graphs with 5 vertices and having no triangles which can occur as prime graphs of some finite groups finally we show that the prime graph of a finite group can never be a cycle nor a tree with at least 5 vertices

nonuniqueness for a parabolic spde with frac 3 4 varepsilon holder diffusion coefficients motivated by girsanov s nonuniqueness examples for sdes we prove nonuniqueness for the parabolic stochastic partial differential equation spde partial derivative u partial derivative t delta 2 u t x vertical bar u t x vertical bar gamma w t x u 0 x 0 here w is a space time white noise on r x r more precisely we show the above stochastic pde has a nonzero solution for 0 partial derivative u partial derivative t delta 2 u t x sigma u t x w t x if a is holder continuous of index gamma 3 4 hence our examples show this result is essentially sharp the situation for the above class of parabolic spde s is therefore similar to their finite dimensional counterparts but with the index 3 4 in place of 1 2 the case gamma 1 2 of the first equation above is particularly interesting as it arises as the scaling limit of the signed mass for a system of annihilating critical branching random walks

on p parts of character degrees and conjugacy class sizes of finite groups abstract let g be a finite group and irr g the set of irreducible complex characters of g let e p g be the largest integer such that p e p g divides 1 for some irr g we show that g f g p p k e p g for a constant k this settles a conjecture of a moreto 13 conjecture 4 we also study the related problems of the p parts of conjugacy class sizes of finite groups

a class of outer generalized inverses abstract in any semigroup or semigroup s it is shown that the moore penrose inverse y a the author s pseudo inverse y a chipman s weighted inverse and the bott duffin inverse are all special cases of the more general class of b c inverses y s satisfying y bsy ysc yab b and cay c these b c inverses always satisfy yay y are always unique when they exist and exist if and only if b scab and c cabs in which case under the partial order m of mitsch y is also the unique m greatest element of the set x a x a b c x x s xax x and x bsx xsc and the unique m least element of z a z a b c z z s zaz z zab b and caz c the above all holds in arbitrary semigroups s hence in particular in any associative ring r for any complex n n matrices a b c an efficient uniform procedure is given to compute the b c inverse of a whenever it exists in the ring case a r is called weakly invertible if there exist b c r satisfying 1 b 1 a r 1 c r 1 a such that a has a b c inverse y satisfying ay ya and it is shown that a is weakly invertible if and only if a is strongly clean in the sense of nicholson i e a u e for some unit u and idempotent e with eu ue

lie type like groups abstract a finite simple group of lie type in defining characteristic p has exactly two p blocks the principal block and a block of defect zero consisting of the steinberg character whose degree is the p part of the order of the group in this paper we characterize finite groups g which have exactly the principal p block and a p block of defect zero consisting of an irreducible character of degree g p

jacobson s lemma for the generalized drazin inverse abstract 0truemm0truemm we study properties of elements in a ring which admit the generalized drazin inverse it is shown that the element 1 ab is generalized drazin invertible if and only if so is 1 ba and a formula for the generalized drazin inverse of 1 ba in terms of the generalized drazin inverse and the spectral idempotent of 1 ab is provided further recent results relating to the drazin index can be recovered from our theorems

integration by parts on the law of the reflecting brownian motion abstract we prove an integration by parts formula on the law of the reflecting brownian motion x b in the positive half line where b is a standard brownian motion in other terms we consider a perturbation of x of the form x e x e h with h smooth deterministic function and e 0 and we differentiate the law of x e at e 0 this infinitesimal perturbation changes drastically the set of zeros of x for any e 0 as a consequence the formula we obtain contains an infinite dimensional generalized functional in the sense of schwartz defined in terms of hida s renormalization of the squared derivative of b and in terms of the local time of x at 0 we also compute the divergence on the wiener space of a class of vector fields not taking values in the cameron martin space