Locally finite Sylp*-groups

Variants of some of the Brauer-Fowler theorems

The enhanced power graph [Formula: see text] of a group [Formula: see text] is the graph with vertex set [Formula: see text] such that two vertices [Formula: see text] and [Formula: see text] are adjacent if they are contained in the same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between undirected power graph of finite groups is an isomorphism between enhanced power graphs of these groups, and we find all finite groups [Formula: see text] for which [Formula: see text] is abelian, all finite groups [Formula: see text] with [Formula: see text] being prime power, and all finite groups [Formula: see text] with [Formula: see text] being square-free. Also, we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly perfect enhanced power graph.

The classifying space BG of a topological group G can be filtered by a sequence of subspaces B(q,G) using the descending central series of free groups. If G is finite, describing them as homotopy colimits is convenient when applying homotopy theoretic methods. In this thesis we introduce natural subspaces B(q,G)_p of B(q,G) defined for a fixed prime p. We show that B(q,G) is stably homotopy equivalent to a wedge sum of B(q,G)_p as p runs over the primes dividing the order of G. Colimits of abelian groups play an important role in understanding the homotopy type of these spaces. Extraspecial p-groups are key examples, for which these colimits turn out to be finite. We prove that for extraspecial p-groups of rank at least 4 the space B(2,G) does not have the homotopy type of a K(π,1) space. Furthermore, we give a group theoretic condition, applicable to symmetric groups and general linear groups, which implies the space B(2,G) not having the homotopy type of a K(π,1) space. For a finite group G, we compute the complex K-theory of B(2,G) modulo torsion.

This thesis records an attempt to prove the two conjecture: Conjecture A: Every finite non-regular primitive permutation group of degree n contains permutations fixing one point but fixing at most $n^{1/2}$ points. Conjecture C: Every finite irreducible linear group of degree m > 1 contains an element whose fixed-point space has dimension at most m/2. Variants of these conjectures are formulated, and C is reduced to a special case of A. The main results of the investigation are: Theorem 2: Every finite non-regular primitive permutation group of degree n contains permutations which fix one point but fix fewer than (n+3)/4 points. Theorem 3: Every finite non-regular primitive soluble permutation group of degree n contains permutations which fix one point but fix fewer than $n^{7/18}$ points. Theorem 4: If H is a finite group, F is a field whose characteristic is 0 or does not divide the order of H, and M is a non-trivial irreducible H-module of dimension m over F, then there is an element h in H whose fixed-point space in M has dimension less than m/2. Theorem 5: If H is a finite soluble group, F is any field, and M is a non-trivial irreducible H-module of dimension m over F, then there is an element h in H whose fixed-point space in M has dimension less than 7m/18. Proofs of these assertions are to be found in Chapter II; examples which show the limitations on possible strenghtenings of the conjectures and results are marshalled in Chapter III. A detailed formulation of the problems and results is contained in section 1.

A subgroup [Formula: see text] of a group [Formula: see text] is said to be an [Formula: see text]-subgroup of [Formula: see text] if [Formula: see text]. We analyze the structure of a finite group [Formula: see text] under the assumption that some given subgroups of [Formula: see text] are [Formula: see text]-subgroups of [Formula: see text]. A new characterization of finite abelian groups and some new criteria for [Formula: see text]-nilpotence and nilpotence of finite groups will be obtained. Moreover, we will obtain two criteria for a finite group to lie in a given solvably saturated formation containing the class of finite supersolvable groups.

In this article, we introduce the idea of Lie regular elements and study 2 × 2 Lie regular matrices. It is shown that the linear groups GL(2, ℤ2 n ), GL(2, ℤ p n ), and GL(2, ℤ2p ) (where p is an odd prime) can be genrated by Lie regular matrices. Presentations of linear groups GL(2, ℤ4), GL(2, ℤ6), GL(2, ℤ8), and GL(2, ℤ10) are also given.

This thesis presents some contributions in additive combinatorics and arithmetic Ramsey theory. More specifically, it deals with the interaction between combinatorics, number theory and additive combinatorics. This area saw a great improvement with the Szemerédi Regularity Lemma and some of the results that followed. The Regularity Lemma and its consequences have become a widely used tool in graph theory, combinatorics and number theory. Furthermore, its language and point of view has deeply changed the face of additive number theory, a fact universally acknowledged by the Abel award given to Szemerédi in 2012. One of the main reasons for the prize has been Szemerédi's theorem, a result regarding the existence of arbitrarily long arithmetic progressions in dense sets of the integers, the proof of which uses the Regularity Lemma in a key step. One of the earlier consequences of the Regularity Lemma was the Removal Lemma for graphs that was used by Ruzsa and Szemerédi to show Roth theorem, regarding the existence of 3-term arithmetic progressions in dense sets of the integers, in a combinatorial way. The Removal Lemma states that in any graph K with few copies of a subgraph, say a triangle, we can remove few edges from K so that the result contains no copy of the subgraph. This has become a key tool in the applications of the so-called Regularity Method, which has extensive literature in combinatorics, graph theory, number theory and computer science. In 2005 Green introduced a regularity lemma for Abelian groups as well as an algebraic removal lemma. The removal lemma for groups states that, for a given finite Abelian group G, if there are o(|G|^3) solution to x+y+z+t=0 with the variables taking values in S, a subset of G, then we can remove o(|G|) elements from S to make the set S solution-free. The main contributions of this work corresponds to extensions of the removal lemma for groups to either more general contexts, like non-necessary Abelian finite groups, or to linear systems of equations for finite Abelian groups. The main goal is to give a comprehensive and more general framework for many results in additive number theory like Szemerédi Theorem. In particular, we show that the removal lemma for groups by Green can be extended to non-necessary Abelian finite groups. Moreover, we prove a removal lemma for linear systems on finite fields: for every e>0 there exists a d>0 such that if A is a (k x m) linear system of equations with coefficients in a finite field F and the number of solutions to Ax=b, where each variable takes values from a subset Si in F is less than d times |F| raised to m-k, then by removing less than e|F| elements in each Si we can make the resulting sets solution-free, thus solving a conjecture by Green to that respect. Even more, if A is an integer linear system, G is a finite Abelian group, and the determinantal of A and |G| are coprime, then a similar statement holds. Let us mention that the last result allows us to characterize those linear systems where any set S with size proportional to G has a nontrivial solution in S, provided |G| is large enough. This extends the validity of Szmerédi's theorem to finite Abelian groups. These extensions of the removal lemma have been used in arithmetic Ramsey theory to obtain counting results for the number of monochromatic solutions of linear systems. The main result from a work by Frankl, Graham and Rödl in '88 states that the number of monochromatic solutions of regular systems in integer intervals is in fact a positive proportion of the total number of solutions. We give analogous results for solutions in Abelian groups with bounded exponent, for which the main tool in the torsion-free case cannot be applied. Density versions of these counting results are also obtained, in this case with a full characterization.

On the extendability of Bi-Cayley graphs of finite abelian groups

Perfect nonlinear functions between two finite abelian groups were studied by Carlet and Ding (J Complex 20:205---244, 2004) and Pott (Discret Math Appl 138:177---193, 2004), which can be regarded as a generalization of bent functions on finite abelian groups studied by Logachev et al. (Discret Math Appl 7:547---564, 1997). Poinsot (J Discret Math Sci Cryptogr 9:349---364, 2006), (Cryptogr Commun 4:1---23, 2012) extended this research to arbitrary finite groups, and characterized bent functions on finite nonabelian groups as well as perfect nonlinear functions between two arbitrary finite groups by the Fourier transforms of the related functions at irreducible unitary representations. The purpose of this paper is to study the characterizations of the bentness (perfect nonlinearity) of functions on arbitrary finite groups by the Fourier transforms of the related functions at irreducible characters. We will also give a characterization of a perfect nonlinear function by the relative pseudo-difference family.

The aim of this note is to offer a summary of the definitions and properties of arithmetic symbols on the linear group Gl(n, F) —F being an arbitrary discrete valuation field—, and to show that the natural generalizations of the Parshin symbol on an algebraic surface S to the linear group Gl(n, ΣS) do not allow us to define new 2-dimensional symbols on S.

This article began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-n, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal nontrivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a $p$-adic vector space associated with $M$. This leads to our second variation, which is a study of the finite linear groups that can arise.

Weights for symmetric and general linear groups

In [3] we initiated our study of the automorphism groups of a certain class of near-rings. Specifically, let P be any complex polynomial and let P denote the near-ring of all continuous selfmaps of the complex plane where addition of functions is pointwise and the product fg of two functions f and g in P is defined by fg=f∘P∘g. The near-ring P is referred to as a laminated near-ring with laminating element P. In [3], we characterised those polynomials P(z)=anzn + an−1zn−1 +…+a0 for which Aut P is a finite group. We are able to show that Aut P is finite if and only if Deg P≧3 and ai ≠ 0 for some i ≠ 0, n. In addition, we were able to completely determine those infinite groups which occur as automorphism groups of the near-rings P. There are exactly three of them. One is GL(2) the full linear group of all real 2×2 nonsingular matrices and the other two are subgroups of GL(2). In this paper, we begin our study of the finite automorphism groups of the near-rings P. We get a result which, in contrast to the situation for the infinite automorphism groups, shows that infinitely many finite groups occur as automorphism groups of the near-rings under consideration. In addition to this and other results, we completely determine Aut P when the coefficients of P are real and Deg P = 3 or 4.

We show that every action of a finite dihedral group on a closed orientable surface F \mathcal F extends to a 3-dimensional handlebody V \mathcal V , with ∂ V = F \partial \mathcal V=\mathcal F . In the case of a finite abelian group G G , we give necessary and sufficient conditions for a G G -action on a surface to extend to a compact 3 3 -manifold, or, equivalently in this case, to a 3-dimensional handlebody; in particular all (fixed-point) free actions of finite abelian groups extend to handlebodies. This is no longer true for free actions of arbitrary finite groups: we give a procedure which allows us to construct free actions of finite groups on surfaces which do not extend to a handlebody. We also show that the unique Hurwitz action of order 84 ( g − 1 ) 84(g-1) of P S L ( 2 , 27 ) PSL(2,27) on a surface F \mathcal F of genus g = 118 g=118 does not extend to any compact 3-manifold M M with ∂ M = F \partial M=\mathcal F , thus resolving the only case of Hurwitz actions of type P S L ( 2 , q ) PSL(2,q) of low order which remained open in an earlier paper (Math. Proc. Cambridge Philos. Soc. 117 (1995), 137–151).

Let G be a finite nonabelian group. Bent functions on G are defined by the Fourier transforms at irreducible representations of G. We introduce a dual basis $${\widehat{G}}$$ , consisting of functions on G determined by its unitary irreducible representations, that will play a role similar to the dual group of a finite abelian group. Then we define the Fourier transforms as functions on $${\widehat{G}}$$ , and obtain characterizations of a bent function by its Fourier transforms (as functions on $${\widehat{G}}$$ ). For a function f from G to another finite group, we define a dual function $${\widetilde{f}}$$ on $${\widehat{G}}$$ , and characterize the nonlinearity of f by its dual function $${\widetilde{f}}$$ . Some known results are direct consequences. Constructions of bent functions and perfect nonlinear functions are also presented.