Maximal Class Research Articles

Monotonicities of quasi‐normed Calderón–Lozanovskiĭ spaces with applications to approximation problems

AbstractWe consider the geometric structure of quasi‐normed Calderón–Lozanovskiĭ spaces. First, we study relations between the quasi‐norm and the quasi‐modular “near zero” and “near one,” which are fundamental for the theory. With their help, we provide a precise description of the basic monotonicity properties. In comparison with the well‐known normed case, we develop a number of new techniques and methods, among which the conditions and play a crucial role. From our general results, we conclude the criteria for monotonicity properties in quasi‐normed Orlicz spaces, which are new even in this particular context. We consider both the function and the sequence case as well as we admit degenerated Orlicz functions, which provides us with a maximal class of spaces under consideration. We also discuss the applications of suitable properties to the best dominated approximation problems in quasi‐Banach lattices.

On the relation of character codegrees and the minimal faithful quasi-permutation representation degree of p-groups

For a finite group G, we denote by c ( G ) , the minimal degree of a faithful representation of G by quasi-permutation matrices over C . For an irreducible character χ of G, the codegree of χ is defined as cod ( χ ) = | G / ker ( χ ) | / χ ( 1 ) . In this article, we establish equality between c ( G ) and a Q ≥ 0 -sum of codegrees of some irreducible characters of a non-abelian p-group G of odd order. We also study the relation between c ( G ) and irreducible character codegrees for various classes of non-abelian p-groups, such as, p-groups with cyclic center, maximal class p-groups, GVZ p-groups, and others.

Beauville p-groups of wild type and groups of maximal class

AbstractLet G be a Beauville p-group. If G exhibits a ‘good behaviour’ with respect to taking powers, then every lift of a Beauville structure of $$G/\Phi (G)$$ G / Φ ( G ) is a Beauville structure of G. We say that G is a Beauville p-group of wild type if this lifting property fails to hold. Our goal in this paper is twofold: firstly, we fully determine the Beauville groups within two large families of p-groups of maximal class, namely metabelian groups and groups with a maximal subgroup of class at most 2; secondly, as a consequence of the previous result, we obtain infinitely many Beauville p-groups of wild type.

On some realizable metabelian 5-groups

Let G be a 5-group of maximal class and G' = [G, G] its derived group. Assume that the abelianization G/G' is of type (5, 5) and the transfers from H1 to G' and from H2 to G' are trivial, where H1 and H2 are two maximal normal subgroups of G. Then G is completely determined with the isomorphism class groups of maximal class. Moreover the group G is realizable with some fields k, which is the normal closure of a pure quintic field.

Representation zeta function of a family of maximal class groups: Non-exceptional primes

Abstract We use a constructive method to obtain all but finitely many 𝑝-local representation zeta functions of a family of finitely generated nilpotent groups M n M_{n} with maximal nilpotency class. For representation dimensions coprime to all primes p < n p<n , we construct all irreducible representations of M n M_{n} by defining a standard form for the matrices of these representations.

Character degrees of 5-groups of maximal class

Abstract Let 𝐺 be a 5-group of maximal class with major centralizer G 1 = C G ⁢ ( G 2 / G 4 ) G_{1}=C_{G}({G_{2}}/{G_{4}}) . In this paper, we prove that the irreducible character degrees of a 5-group 𝐺 of maximal class are almost determined by the irreducible character degrees of the major centralizer G 1 G_{1} and show that the set of irreducible character degrees of a 5-group of maximal class is either { 1 , 5 , 5 3 } \{1,5,5^{3}\} or { 1 , 5 , … , 5 k } \{1,5,\ldots,5^{k}\} with k ≥ 1 k\geq 1 .

Proving a conjecture for fusion systems on a class of groups

We prove the conjecture that exotic and block-exotic fusion systems coincide holds for all fusion systems on exceptional p-groups of maximal nilpotency class, where p≥5. This is done by considering a family of exotic fusion systems discovered by Parker and Stroth. Together with a previous result by the author, which we also generalise in this paper, and a result by Grazian and Parker this implies the conjecture for fusion systems on such groups. Considering small primes, there are no exotic fusion systems on 2-groups of maximal class and for p=3, we prove block-exoticity of two exotic fusion systems described by Diaz–Ruiz–Viruel.

Biderivations, commuting mappings and 2-local derivations of -graded Lie algebras of maximal class

In Fialowski's classification for algebras of maximal class, there are three Lie algebras of maximal class with one-dimensional homogeneous components: m 0 , L 1 and m 2 . In this paper, we study their biderivations by considering the embedded mapping to derivation algebras. Then we determine commuting mappings on these algebras as an application of biderivations. Finally, local and 2-local derivations for these three algebras are characterized as the given gradings.

Bounds on the order of the Schur multiplier of p-groups

In 1956, Green provided a bound on the order of the Schur multiplier of p-groups. This bound, given as a function of the order of the group, is the best possible. Since then, the bound has been refined numerous times by adding other inputs to the function, such as, the minimal number of generators of the group and the order of the derived subgroup. We strengthen these bounds by adding another input, the group's nilpotency class. The specific cases of nilpotency class 2 and maximal class are discussed in greater detail.

Character degrees of normally monomial 𝑝-groups of maximal class

AbstractA finite group 𝐺 is normally monomial if all its irreducible characters are induced from linear characters of normal subgroups of 𝐺. In this paper, we study the largest irreducible character degree and the maximal abelian normal subgroup of normally monomial 𝑝-groups of maximal class in terms of 𝑝. In particular, we determine all possible irreducible character degree sets of normally monomial 5-groups of maximal class.

Completeness, closedness and metric reflections of pseudometric spaces

It is well-known that a metric space (X,d) is complete iff the set X is closed in every metric superspace of (X,d). For a given pseudometric space (Y,ρ), we describe the maximal class CEC(Y,ρ) of superspaces of (Y,ρ) such that (Y,ρ) is complete if and only if Y is closed in every (Z,Δ)∈CEC(Y,ρ).We also introduce the concept of pseudoisometric spaces and prove that spaces are pseudoisometric iff their metric reflections are isometric. The last result implies that a pseudometric space is complete if and only if this space is pseudoisometric to a complete pseudometric space.

Strata of p‐origamis

AbstractGiven a two‐generated group of prime‐power order, we investigate the singularities of origamis whose deck group acts transitively and is isomorphic to the given group. Geometric and group‐theoretic ideas are used to classify the possible strata, depending on the prime‐power order. We then show that for many interesting known families of two‐generated groups of prime‐power order, including all regular, or powerful ones, or those of maximal class, each group admits only one possible stratum. However, we also construct examples of two‐generated groups of prime‐power order, which do not determine a unique stratum.

On the Schur multiplier of finite 𝑝-groups of maximal class

Abstract In this article, we prove that the Schur multiplier of a finite 𝑝-group of maximal class of order p n p^{n} ( 4 ≤ n ≤ p + 1 4\leq n\leq p+1 ) is elementary abelian. The case n = p + 1 n=p+1 settles a question raised by Primož Moravec in an earlier article.

Thin subalgebras of Lie algebras of maximal class

For every field $$\mathbb{F}$$ which has a quadratic extension $$\mathbb{E}$$ we show there are non-metabelian infinite-dimensional thin graded Lie algebras all of whose homogeneous components, except the second one, have dimension 2. We construct such Lie algebras as $$\mathbb{F}$$ -subalgebras of Lie algebras M of maximal class over $$\mathbb{E}$$ . We characterise the thin Lie $$\mathbb{F}$$ -subalgebras of M generated in degree 1. Moreover, we show that every thin Lie algebra L whose ring of graded endomorphisms of degree zero of L3 is a quadratic extension of $$\mathbb{F}$$ can be obtained in this way. We also characterise the 2-generator $$\mathbb{F}$$ -subalgebras of a Lie algebra of maximal class over $$\mathbb{E}$$ which are ideally r-constrained for a positive integer r.

Some finite p-groups whose normal subgroups are characteristic

In this article, we classify some finite p-groups whose normal subgroups are characteristic. Particularly we restrict our attention to all finite p-groups of maximal class with an abelian maximal subgroup, finite 3-groups of maximal class, finite split metacyclic p-groups, minimal non-abelian p-groups and all p-groups of maximal class of order less than p 6.

Some necessary and sufficient conditions on commuting automorphisms of some finite p-groups

Let G be a finite non-abelian p-group of order greater than p 4, p an odd prime, such that is cyclic and Z(H) is elementary abelian, where We prove that the set of all commuting automorphisms of G forms a subgroup of if and only if is abelian. Also, we find the structure of for a finite 2-group G of almost maximal class with cyclic center Z(G), where denotes the set of all central automorphisms of G.

The Cauchy problem for the fast p-Laplacian evolution equation. Characterization of the global Harnack principle and fine asymptotic behaviour

We study fine global properties of nonnegative solutions to the Cauchy Problem for the fast p-Laplacian evolution equation ut=Δpu on the whole Euclidean space, in the so-called “good fast diffusion range” 2NN+1<p<2. It is well-known that non-negative solutions behave for large times as B, the Barenblatt (or fundamental) solution, which has an explicit expression.We prove the so-called Global Harnack Principle (GHP), that is, precise global pointwise upper and lower estimates of nonnegative solutions in terms of B. This can be considered the nonlinear counterpart of the celebrated Gaussian estimates for the linear heat equation.We characterize the maximal (hence optimal) class of initial data such that the GHP holds, by means of an integral tail condition, easy to check. The GHP is then used as a tool to analyze the fine asymptotic behaviour for large times. For initial data that satisfy the same integral condition, we prove that the corresponding solutions behave like the Barenblatt with the same mass, uniformly in relative error.When the integral tail condition is not satisfied we show that both the GHP and the uniform convergence in relative error, do not hold anymore, and we provide also explicit counterexamples. We then prove a “generalized GHP”, that is, pointwise upper and lower bounds in terms of explicit profiles with a tail different from B. Finally, we derive sharp global quantitative upper bounds of the modulus of the gradient of the solution, and, when data are radially decreasing, we show uniform convergence in relative error for the gradients.To the best of our knowledge, analogous issues for the linear heat equation p=2, do not possess such clear answers, only partial results are known.

The Groups and Nilpotent Lie Rings of Order p 8 with Maximal Class

We classify and count the nilpotent Lie rings of order p 8 with maximal class for p ≥ 5 . This also provides a classification of the groups of order p 8 with maximal class for p ≥ 11 via the Lazard correspondence. We also record the number of nilpotent Lie rings/groups of order pn with maximal class for n ≤ 7 from currently known data and discuss its asymptotic behavior as n grows and its potential connection to Higman’s PORC conjecture.

A family of finitep‐groups satisfying Carlson's depth conjecture

AbstractLet be a prime number and letrbe an integer with . For eachr, let moreover denote the unique quotient of the maximal class pro‐pgroup of size . We show that the mod‐pcohomology ring of has depth one and that, in turn, it satisfies the equalities in Carlson's depth conjecture [2]. This is the first family of finitep‐groups for which Carlson's depth conjecture has been verified besidesp‐groups of abelian type mod‐pcohomology or extraspecialp‐groups. Moreover, this computation is possible without first describing the structure of the cohomology ring.

Galois trees in the graph of p-groups of maximal class

The investigation of the graph Gp associated with the finite p-groups of maximal class was initiated by Blackburn (1958) and became a deep and interesting research topic since then. Leedham-Green & McKay (1976–1984) introduced skeletons of Gp, described their importance for the structural investigation of Gp and exhibited their relation to algebraic number theory. Here we go one step further: we partition the skeletons into so-called Galois trees and study their general shape. In the special case p⩾7 and p≡5mod6, we show that they have a significant impact on the periodic patterns of Gp conjectured by Eick, Leedham-Green, Newman & O'Brien (2013). In particular, we use Galois trees to prove a conjecture by Dietrich (2010) on these periodic patterns.