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A basic fact taught in undergraduate algebra courses is that every finite nilpotent group is a direct product of $p$-groups. Already Bruck observed that this does not generalize to loops. In particular, there exist nilpotent loops of size $6$ which are not direct products of loops of size $2$ and $3$. Still we show that for every finite nilpotent loop $(A,\cdot)$ there exists a binary term operation $*$ such that $(A,*)$ is a direct product of nilpotent loops of prime power order, i.e., $(A,*)$ is supernilpotent.

We study loops which are universal (that is, isotopically invariant) with respect to the property of flexibility ($xy\cdot x = x\cdot yx$). We also weaken this to semi-universality, that is, loops in which every left and right isotope is flexible, but not necessarily every isotope. One of our main results is that universally flexible, inverse property loops are Moufang loops. On the other hand, semi-universally flexible, inverse property loops are diassociative. We also examine the relationship between universally flexible loops and middle Bol loops. The paper concludes with some open problems., Comment: 13pp

We study abelian-by-cyclic Moufang loops. We construct all split $3$-divisible abelian-by-cyclic Moufang loops from so-called Moufang permutations on abelian groups $(X,+)$, which are permutations that deviate from an automorphism of $(X,+)$ by an alternating biadditive mapping (satisfying certain properties). More generally, we obtain additional abelian-by-cyclic Moufang loops from so-called construction pairs. As an aside, we show that in the Moufang loops $Q$ obtained from a construction pair on $(X,+)$ the abelian normal subgroup $(X,+)$ induces an abelian congruence of $Q$ if and only if $Q$ is a group.

We prove that a normal subloop $X$ of a Moufang loop $Q$ induces an abelian congruence of $Q$ if and only if each inner mapping of $Q$ restricts to an automorphism of $X$ and $u(xy) = (uy)x$ for all $x,y\in X$ and $u\in Q$. The former condition can be omitted when $X$ is $3$-divisible. This characterization is then used to show that classically solvable finite $3$-divisible Moufang loops are congruence solvable.

In groups, an abelian normal subgroup induces an abelian congruence. We construct a class of centrally nilpotent Moufang loops containing an abelian normal subloop that does not induce an abelian congruence. On the other hand, we prove that in $6$-divisible Moufang loops, every abelian normal subloop induces an abelian congruence. In loops, congruence solvability adopted from the universal-algebraic commutator theory of congruence modular varieties is strictly stronger than classical solvability adopted from group theory. It is an open problem whether the two notions of solvability coincide in Moufang loops. We prove that they coincide in $6$-divisible Moufang loops and in Moufang loops of odd order. In fact, we show that every Moufang loop of odd order is congruence solvable, thus strengthening Glauberman's Odd Order Theorem for Moufang loops., Comment: Minor revision. Includes a shorter proof of Theorem 1.11 for the case p=3. To appear in Journal of Algebra

The gyrogroup is the closest algebraic structure to the group ever discovered. It has a binary operation $\star$ containing an identity element such that each element has an inverse. Furthermore, for each pair $(a,b)$ of elements of this structure there exists an automorphism $\gyr{a,b}{}$ with this property that left associativity and left loop property are satisfied. Since each gyrogroup is a left Bol loop, some results of Burn imply that all gyrogroups of orders $p, 2p$ and $p^2$ are groups. The aim of this paper is to classify gyrogroups of orders 8, 12, 15, 18, 20, 21, and 28., Comment: 9 pages

We prolong research of Schroder quasigroups and Schroder T-quasigroups., Comment: 13 pages

Automorphic loops are loops in which all inner mappings are automorphisms. This variety of loops includes groups and commutative Moufang loops. A half-isomorphism $f : G \longrightarrow K$ between multiplicative systems $G$ and $K$ is a bijection from $G$ onto $K$ such that $f(ab)\in\{f(a)f(b), f(b)f(a)\}$ for any $a,b\in G$. A half-isomorphism is trivial when it is either an isomorphism or an anti-isomorphism. Consider the class of automorphic loops such that the equation $x\cdot(x\cdot y) = (y\cdot x)\cdot x $ is equivalent to $x\cdot y = y\cdot x$. Here we show that this class of loops includes automorphic loops of odd order and uniquely $2$-divisible. Furthermore, we prove that every half-isomorphism between loops in that class is trivial., Comment: 15 pages

A Bol loop is a loop that satisfies the identity $x((yz)y)=((xy)z)y$. In this paper, we give a construction of the free Bol loops of exponent two. We define a canonical form of all their elements and describe their multiplication law based on this form., Comment: 16 pages

We proceed the research of generalized quasigroup derivatives started in early papers of the last co-author ([20, p. 212], [13]). For any quasigroup there exist 648 generalized derivatives. Here we study the problem about existence of units (left, right, middle) in quasigroups that are a generalized derivative of a quasigroup. There exist 1944 various cases. For every case we find a proof or a counterexample, often using Prover or Mace [15, 14]., Comment: 16 pages

Given a uniquely 2-divisible group $G$, we study a commutative loop $(G,\circ)$ which arises as a result of a construction in \cite{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 \cite{greer14}, it is conjectured that $G$ is metabelian if and only if $(G, \circ)$ is an automorphic loop. We answer a portion of this conjecture in the affirmative: in particular, we show that if $G$ is a split metabelian group of odd order, then $(G, \circ)$ is automorphic.

We prolong Kunen research about existence of units (left, right, two-sided) in quasigroups with classical Bol-Moufang type identities. These identities were listed in Fenvesh article., Comment: 23 pages

We present results about groupoids of small order with Bol-Moufang type identities both classical and non-classical which are listed in [7, 8]., Comment: 10 pages

Let $Q$ be a loop. If $S\leq Q$ is such that $\varphi(S) \subseteq S$ for each standard generator of $\mathrm{Inn}(Q)$, then $S$ does not have to be a normal subloop. In an LC loop the left and middle nucleus coincide and form a normal subloop. The identities of Osborn loops are obtained by applying the idea of nuclear identification, and various connections of Osborn loops to Moufang and CC loops are discussed. Every Osborn loop possesses a normal nucleus, and this nucleus coincides with the left, the right and the middle nucleus. Loops that are both Buchsteiner and Osborn are characterized as loops in which each square is in the nucleus., Comment: 16 pages; v.2: typos fixed; bibliography entry fixed

A bijection $f$ of a loop $L$ is a half-automorphism if $f(xy)\in \{f(x)f(y),f(y)f(x)\}$, for any $x,y\in L$. A half-automorphism is nontrivial when it is neither an automorphism nor an anti-automorphism. A Chein loop $L=G\cup Gu$ is a Moufang loop constructed from a group $G$ and an element $u$ of order $2$ outside of $G$. In this paper, the half-automorphism group of finite Chein loops is described., Comment: 9 pages

The aim of our paper is to generalize the tangent prolongation of Lie groups to non-associative multiplications and to examine how the weak associative and weak inverse properties are transferred to the multiplication defined on the tangent bundle. We obtain that the tangent prolongation of a $\mathcal{C}^r$-differentiable loop ($r\geq 1$) is a $\mathcal{C}^{r-1}$-differentiable loop that acquires the classical weak inverse and weak associative properties of the initial loop.

Our paper deals with the investigation of extensions of commutative groups by loops so that the quasigroups that result in the multiplication between cosets of the kernel subgroup are T-quasigroups. We limit our study to extensions in which the quasigroups determining the multiplication are linear functions without constant term, called linear abelian extensions. We characterize constructively such extensions with left-, right-, or inverse properties using a general construction according to an equivariant group action principle. We show that the obtained constructions can be simplified for ordered loops. Finally, we apply our characterization to determine the possible cardinalities of the component loop of finite linear abelian extensions.

For a unital ring $S$, an $S$-linear quasigroup is a unital $S$-module, with automorphisms $\rho$ and $\lambda$ giving a (nonassociative) multiplication $x\cdot y=x^\rho+y^\lambda$. If $S$ is the field of complex numbers, then ordinary characters provide a complete linear isomorphism invariant for finite-dimensional $S$-linear quasigroups. Over other rings, it is an open problem to determine tractably computable isomorphism invariants. The paper investigates this isomorphism problem for $\mathbb{Z}$-linear quasigroups. We consider the extent to which ordinary characters classify $\mathbb{Z}$-linear quasigroups and their representations of the free group on two generators. We exhibit non-isomorphic $\mathbb{Z}$-linear quasigroups with the same ordinary character. For a subclass of $\mathbb{Z}$-linear quasigroups, equivalences of the corresponding ordinary representations are realized by permutational intertwinings. This leads to a new equivalence relation on $\mathbb{Z}$-linear quasigroups, namely permutational similarity. Like the earlier concept of central isotopy, permutational similarity is intermediate between isomorphism and isotopy., Comment: J. Math. Sci. (2019)

Any Neumann quasigroup $(Q, \cdot)$ (quasigroup with Neumann identity $ x \cdot(yz \cdot yx) = z$ is called Neumann quasigroup) can be presented in the form $x\cdot y = x-y$, where $(Q, +)$ is an abelian group. Automorphism group of Neumann quasigroup coincides with the group $Aut(Q, +)$. Any Schweizer quasigroup (quasigroup with Schweizer identity $xy \cdot xz = zy$ is called Schweizer quasigroup) is a Neumann quasigroup and vice versa. Any Neumann quasigroup is a GA-quasigroup., Comment: 7 pages

Fix a parameter $K\geqslant 1$. A $K$-approximate subgroup is a finite set $A$ in a group $G$ which contains the identity, is symmetric and such that $A.A$ can covered by $K$ left translates of $A$. This article deals with the generalisation of the concept of approximate groups in the case of loops which we call approximate loops and the description of $K$-approximate subloops when the ambient loop is a finitely generated commutative moufang loop. Specifically we have a Freiman type theorem where such approximate subloops are controlled by arithmetic progressions defined in the commutative moufang loops.

Groupoid $(Q, \cdot)$ in which equality $(xy)Jx = y$ is true for all $x, y \in Q$, where $J$ is a map of the set $Q$, is a CI-quasigroup., Comment: 3 pages

We give a general construction for right conjugacy closed loops, using $GL(2,q)$ for $q$ a prime power. Under certain conditions, the loops constructed are simple, giving the first general construction for finite, simple right conjugacy closed loops. We give a complete description of the isomorphism classes for the construction, yielding an exact count of non isomorphic loops for each $q$.

Right Bol loops are loops satisfying the identity $((zx)y)x = z((xy)x)$, and right Bruck loops are right Bol loops satisfying the identity $(xy)^{-1} = x^{-1}y^{-1}$. Let $p$ and $q$ be odd primes such that $p>q$. Advancing the research program of Niederreiter and Robinson from $1981$, we classify right Bol loops of order $pq$. When $q$ does not divide $p^2-1$, the only right Bol loop of order $pq$ is the cyclic group of order $pq$. When $q$ divides $p^2-1$, there are precisely $(p-q+4)/2$ right Bol loops of order $pq$ up to isomorphism, including a unique nonassociative right Bruck loop $B_{p,q}$ of order $pq$. Let $Q$ be a nonassociative right Bol loop of order $pq$. We prove that the right nucleus of $Q$ is trivial, the left nucleus of $Q$ is normal and is equal to the unique subloop of order $p$ in $Q$, and the right multiplication group of $Q$ has order $p^2q$ or $p^3q$. When $Q=B_{p,q}$, the right multiplication group of $Q$ is isomorphic to the semidirect product of $\mathbb{Z}_p\times \mathbb{Z}_p$ with $\mathbb{Z}_q$. Finally, we offer computational results as to the number of right Bol loops of order $pq$ up to isotopy., Comment: 23 pages; to appear in J. Algebra

In this note we introduce the concept of an almost left automorphic Moufang loop and study the properties of tangent algebras of smooth loops of this class.

In this article we gave the notion of particular type of functions on a group termed as class assigned functions . Using class assigned functions, we have constructed several right gyrogroup structures on a given non-abelian group.

Andrew Rajah posed at the Loops'11 Conference in Trest, Czech Republic, the following conjecture: Is every variety of loops that satisfies Moufang's theorem contained in the variety of Moufang loops? This paper is motivated by that problem. We give a partial answer to this question and present two types of Steiner loops, one that satisfies Moufang's theorem and another that does not, and neither is Moufang loop.

Groups with commuting inner mappings are of nilpotency class at most two, but there exist loops with commuting inner mappings and of nilpotency class higher than two, called loops of Cs\"org\H{o} type. In order to obtain small loops of Cs\"org\H{o} type, we expand our programme from `Explicit constructions of loops with commuting inner mappings', European J. Combin. 29 (2008), 1662-1681, and analyze the following setup in groups: Let $G$ be a group, $Z\le Z(G)$, and suppose that $\delta:G/Z\times G/Z\to Z$ satisfies $\delta(x,x)=1$, $\delta(x,y)=\delta(y,x)^{-1}$, $z^{yx}\delta([z,y],x) = z^{xy}\delta([z,x],y)$ for every $x$, $y$, $z\in G$, and $\delta(xy,z) = \delta(x,z)\delta(y,z)$ whenever $\{x,y,z\}\cap G'$ is not empty. Then there is $\mu:G/Z\times G/Z\to Z$ with $\delta(x,y) = \mu(x,y)\mu(y,x)^{-1}$ such that the multiplication $x*y=xy\mu(x,y)$ defines a loop with commuting inner mappings, and this loop is of Cs\"org\H{o} type (of nilpotency class three) if and only if $g(x,y,z) = \delta([x,y],z)\delta([y,z],x)\delta([z,x],y)$ is nontrivial. Moreover, $G$ has nilpotency class at most three, and if $g$ is nontrivial then $|G|\ge 128$, $|G|$ is even, and $g$ induces a trilinear alternating form. We describe all nontrivial setups $(G,Z,\delta)$ with $|G|=128$. This allows us to construct for the first time a loop of Cs\"org\H{o} type with an inner mapping group that is not elementary abelian.

In 2004, Cs\"{o}rg\H{o} constructed a loop of nilpotency class three with abelian group of inner mappings. Until now, no other examples were known. We construct many such loops from groups of nilpotency class two by replacing the product $xy$ with $xyh$ in certain positions, where $h$ is a central involution. The location of the replacements is ultimately governed by a symmetric trilinear alternating form.

We investigate the relation between the structure of a Moufang loop and its inner mapping group. Moufang loops of odd order with commuting inner mappings have nilpotency class at most two. $6$-divisible Moufang loops with commuting inner mappings have nilpotency class at most two. There is a Moufang loop of order $2^{14}$ with commuting inner mappings and of nilpotency class three.

We classify Moufang loops of order 64 and 81 up to isomorphism, using a linear algebraic approach to central loop extensions. In addition to the 267 groups of order 64, there are 4262 nonassociative Moufang loops of order 64. In addition to the $15$ groups of order $81$, there are $5$ nonassociative Moufang loops of order $81$, $2$ of which are commutative.