Search results for "combinatoric"
showing 10 items of 1776 documents
The coordinatization of affine planes by rings
1996
With every unitary free module of rank 2 there is naturally associated a generalized affine plane (e.g. the lines are just the cosets of all nonzero 1-generated submodules). Here we solve the converse problem by coordinatizing a given generalized affine plane which satisfies certain versions of Desargues' postulate.
An improvement of a bound of Green
2012
A p-group G of order pn (p prime, n ≥ 1) satisfies a classic Green's bound log p |M(G)| ≤ ½n(n - 1) on the order of the Schur multiplier M(G) of G. Ellis and Wiegold sharpened this restriction, proving that log p |M(G)| ≤ ½(d - 1)(n + m), where |G′| = pm(m ≥ 1) and d is the minimal number of generators of G. The first author has recently shown that log p |M(G)| ≤ ½(n + m - 2)(n - m - 1) + 1, improving not only Green's bound, but several other inequalities on |M(G)| in literature. Our main results deal with estimations with respect to the bound of Ellis and Wiegold.
Additivity of affine designs
2020
We show that any affine block design $$\mathcal{D}=(\mathcal{P},\mathcal{B})$$ is a subset of a suitable commutative group $${\mathfrak {G}}_\mathcal{D},$$ with the property that a k-subset of $$\mathcal{P}$$ is a block of $$\mathcal{D}$$ if and only if its k elements sum up to zero. As a consequence, the group of automorphisms of any affine design $$\mathcal{D}$$ is the group of automorphisms of $${\mathfrak {G}}_\mathcal{D}$$ that leave $$\mathcal P$$ invariant. Whenever k is a prime p, $${\mathfrak {G}}_\mathcal{D}$$ is an elementary abelian p-group.
Free sequences and the tightness of pseudoradial spaces
2019
Let F(X) be the supremum of cardinalities of free sequences in X. We prove that the radial character of every Lindelof Hausdorff almost radial space X and the set-tightness of every Lindelof Hausdorff space are always bounded above by F(X). We then improve a result of Dow, Juhasz, Soukup, Szentmiklossy and Weiss by proving that if X is a Lindelof Hausdorff space, and $$X_\delta $$ denotes the $$G_\delta $$ topology on X then $$t(X_\delta ) \le 2^{t(X)}$$ . Finally, we exploit this to prove that if X is a Lindelof Hausdorff pseudoradial space then $$F(X_\delta ) \le 2^{F(X)}$$ .
Characters, bilinear forms and solvable groups
2016
Abstract We prove a number of results about the ordinary and Brauer characters of finite solvable groups in characteristic 2, by defining and using the concept of the extended nucleus of a real irreducible character. In particular we show that the Isaacs canonical lift of a real irreducible Brauer character has Frobenius–Schur indicator +1. We also show that the principal indecomposable module corresponding to a real irreducible Brauer character affords a quadratic geometry if and only if each extended nucleus is a split extension of a nucleus.
Cardinal estimates involving the weak Lindelöf game
2021
AbstractWe show that if X is a first-countable Urysohn space where player II has a winning strategy in the game $$G^{\omega _1}_1({\mathcal {O}}, {\mathcal {O}}_D)$$ G 1 ω 1 ( O , O D ) (the weak Lindelöf game of length $$\omega _1$$ ω 1 ) then X has cardinality at most continuum. This may be considered a partial answer to an old question of Bell, Ginsburg and Woods. It is also the best result of this kind since there are Hausdorff first-countable spaces of arbitrarily large cardinality where player II has a winning strategy even in the weak Lindelöf game of countable length. We also tackle the problem of finding a bound on the cardinality of a first-countable space where player II has a wi…
Finitary shadows of compact subgroups of $$S(\omega )$$
2020
AbstractLet LF be the lattice of all subgroups of the group $$SF(\omega )$$SF(ω) of all finitary permutations of the set of natural numbers. We consider subgroups of $$SF(\omega )$$SF(ω) of the form $$C\cap SF(\omega )$$C∩SF(ω), where C is a compact subgroup of the group of all permutations. In particular, we study their distribution among elements of LF. We measure this using natural relations of orthogonality and almost containedness. We also study complexity of the corresponding families of compact subgroups of $$S(\omega )$$S(ω).
On a paper of Beltrán and Shao about coprime action
2020
Abstract Assume that A and G are finite groups of coprime orders such that A acts on G via automorphisms. Let p be a prime. The following coprime action version of a well-known theorem of Ito about the structure of a minimal non-p-nilpotent groups is proved: if every maximal A-invariant subgroup of G is p-nilpotent, then G is p-soluble. If, moreover, G is not p-nilpotent, then G must be soluble. Some earlier results about coprime action are consequences of this theorem.
Computing the ℤ2-Cocharacter of 3 × 3 Matrices of Odd Degree
2013
Let F be a field of characteristic 0 and A = M 2, 1(F) the algebra of 3 × 3 matrices over F endowed with the only non trivial ℤ2-grading. Aver'yanov in [1] determined a set of generators for the T 2-ideal of graded identities of A. Here we study the identities in variables of homogeneous degree 1 via the representation theory of the symmetric group, and we determine the decomposition of the corresponding character into irreducibles.
Correspondences of Brauer characters and Sylow subgroup normalizers
2021
Abstract Let p > 3 and q ≠ p be primes, let G be a finite q-solvable group and let P ∈ Syl p ( G ) . Then G has a unique irreducible q-Brauer character of p ′ -degree lying over 1 P if and only if N G ( P ) / P is a q-group. One direction of this result follows from a natural McKay bijection of p ′ -degree irreducible q-Brauer characters, which is obtained under suitable conditions.