Nonassociative triples in involutory loops and in loops of small order

Aleš Drápal; Jan Hora

Displaying similar documents to “Nonassociative triples in involutory loops and in loops of small order”

Automorphic loops and metabelian groups

Mark Greer, Lee Raney (2020)

Commentationes Mathematicae Universitatis Carolinae

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Given a uniquely 2-divisible group $G$ , we study a commutative loop $(G, \circ)$ which arises as a result of a construction in “Engelsche elemente noetherscher gruppen” (1957) by R. Baer. We investigate some general properties and applications of “ $\circ$ ” and determine a necessary and sufficient condition on $G$ in order for $(G, \circ)$ to be Moufang. In “A class of loops categorically isomorphic to Bruck loops of odd order” (2014) by M. Greer, it is conjectured that $G$ is metabelian if and only if $(G, \circ)$ is an automorphic...

On dicyclic groups as inner mapping groups of finite loops

Emma Leppälä, Markku Niemenmaa (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $G$ be a finite group with a dicyclic subgroup $H$ . We show that if there exist $H$ -connected transversals in $G$ , then $G$ is a solvable group. We apply this result to loop theory and show that if the inner mapping group $I (Q)$ of a finite loop $Q$ is dicyclic, then $Q$ is a solvable loop. We also discuss a more general solvability criterion in the case where $I (Q)$ is a certain type of a direct product.

The centre of a Steiner loop and the maxi-Pasch problem

Andrew R. Kozlik (2020)

Commentationes Mathematicae Universitatis Carolinae

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A binary operation “ $\cdot$ ” which satisfies the identities $x \cdot e = x$ , $x \cdot x = e$ , $(x \cdot y) \cdot x = y$ and $x \cdot y = y \cdot x$ is called a Steiner loop. This paper revisits the proof of the necessary and sufficient conditions for the existence of a Steiner loop of order $n$ with centre of order $m$ and discusses the connection of this problem to the question of the maximum number of Pasch configurations which can occur in a Steiner triple system (STS) of a given order. An STS which attains this maximum for a given order is said to be . We show that...

Linear operator identities in quasigroups

Reza Akhtar (2022)

Commentationes Mathematicae Universitatis Carolinae

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We study identities of the form $L_{x_{0}} ϕ_{1} \dots ϕ_{n} R_{x_{n + 1}} = R_{x_{n + 1}} ϕ_{σ (1)} \dots ϕ_{σ (n)} L_{x_{0}}$ in quasigroups, where $n \geq 1$ , $σ$ is a permutation of ${1, ..., n}$ , and for each $i$ , $ϕ_{i}$ is either $L_{x_{i}}$ or $R_{x_{i}}$ . We prove that in a quasigroup, every such identity implies commutativity. Moreover, if $σ$ is chosen randomly and uniformly, it also satisfies associativity with probability approaching $1$ as $n \to \infty$ .

Moufang loops of order coprime to three that cyclically extend groups of dihedral type

Aleš Drápal (2016)

Commentationes Mathematicae Universitatis Carolinae

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This paper completely solves the isomorphism problem for Moufang loops $Q = G C$ where $G ⊴ Q$ is a noncommutative group with cyclic subgroup of index two and $| Z (G) | \leq 2$ , $C$ is cyclic, $G \cap C = 1$ , and $Q$ is finite of order coprime to three.

Classification of quasigroups according to directions of translations II

Fedir Sokhatsky, Alla Lutsenko (2021)

Commentationes Mathematicae Universitatis Carolinae

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In each quasigroup $Q$ there are defined six types of translations: the left, right and middle translations and their inverses. Two translations may coincide as permutations of $Q$ , and yet be different when considered upon the web of the quasigroup. We shall call each of the translation types a direction and will associate it with one of the elements $ι, l, r, s, l s$ and $r s$ , i.e., the elements of a symmetric group $S_{3}$ . Properties of the directions are considered in part 1 of “Classification of quasigroups...

Antiflexible Latin directed triple systems

Andrew R. Kozlik (2015)

Commentationes Mathematicae Universitatis Carolinae

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It is well known that given a Steiner triple system one can define a quasigroup operation $\cdot$ upon its base set by assigning $x \cdot x = x$ for all $x$ and $x \cdot y = z$ , where $z$ is the third point in the block containing the pair ${x, y}$ . The same can be done for Mendelsohn triple systems, where $(x, y)$ is considered to be ordered. But this is not necessarily the case for directed triple systems. However there do exist directed triple systems, which induce a quasigroup under this operation and these are called Latin directed...

Generating varieties for the triple loop space of classical Lie groups

Yasuhiko Kamiyama (2003)

Fundamenta Mathematicae

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For G = SU(n), Sp(n) or Spin(n), let $C_{G} (S U (2))$ be the centralizer of a certain SU(2) in G. We have a natural map $J : G / C_{G} (S U (2)) \to Ω ₀ ³ G$ . For a generator α of $H ⁎ (G / C_{G} (S U (2)); ℤ / 2)$ , we describe J⁎(α). In particular, it is proved that $J ⁎ : H ⁎ (G / C_{G} (S U (2)); ℤ / 2) \to H ⁎ (Ω ₀ ³ G; ℤ / 2)$ is injective.

Differences of two semiconvex functions on the real line

Václav Kryštof, Luděk Zajíček (2016)

Commentationes Mathematicae Universitatis Carolinae

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It is proved that real functions on $ℝ$ which can be represented as the difference of two semiconvex functions with a general modulus (or of two lower $C^{1}$ -functions, or of two strongly paraconvex functions) coincide with semismooth functions on $ℝ$ (i.e. those locally Lipschitz functions on $ℝ$ for which $f_{+}^{'} (x) = {lim}_{t \to x +} f_{+}^{'} (t)$ and $f_{-}^{'} (x) = {lim}_{t \to x -} f_{-}^{'} (t)$ for each $x$ ). Further, for each modulus $ω$ , we characterize the class $D S C_{ω}$ of functions on $ℝ$ which can be written as $f = g - h$ , where $g$ and $h$ are semiconvex with modulus $C ω$ (for some $C > 0$ ) using a new...

A new characterization of symmetric group by NSE

Azam Babai, Zeinab Akhlaghi (2017)

Czechoslovak Mathematical Journal

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Let $G$ be a group and $ω (G)$ be the set of element orders of $G$ . Let $k \in ω (G)$ and $m_{k} (G)$ be the number of elements of order $k$ in $G$ . Let nse $(G) = {m_{k} (G) : k \in ω (G)}$ . Assume $r$ is a prime number and let $G$ be a group such that nse $(G) =$ nse $(S_{r})$ , where $S_{r}$ is the symmetric group of degree $r$ . In this paper we prove that $G ≅ S_{r}$ , if $r$ divides the order of $G$ and $r^{2}$ does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.