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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

Let $\mathbb{F}$ be a finite field of odd order and $a,b\in\mathbb{F}\setminus\{0,1\}$ be such that $\chi(a) = \chi(b)$ and $\chi(1-a)=\chi(1-b)$, where $\chi$ is the extended quadratic character. Let $Q_{a,b}$ be the quasigroup upon $\mathbb{F}$ defined by $(x,y)\mapsto x+a(y-x)$ if $\chi(y-x) \ge 0$, and $(x,y)\mapsto x+b(y-x)$ if $\chi(y-x) = -1$. We show that $Q_{a,b} \cong Q_{c,d}$ if and only if $\{a,b\}= \{\alpha(c),\alpha(d)\}$ for some $\alpha\in \textrm{aut}(\mathbb{F})$. We also characterise $\textrm{aut}(Q_{a,b})$ and exhibit further properties, including establishing when $Q_{a,b}$ is a Steiner quasigroup or is commutative, entropic, left or right distributive, flexible or semisymmetric. In proving our results we also characterise the minimal subquasigroups of $Q_{a,b}$.

A loop $X$ is said to satisfy Moufang's theorem if for every $x,y,z\in X$ such that $x(yz)=(xy)z$ the subloop generated by $x$, $y$, $z$ is a group. We prove that the variety $V$ of Steiner loops satisfying the identity $(xz)(((xy)z)(yz)) = ((xz)((xy)z))(yz)$ is not contained in the variety of Moufang loops, yet every loop in $V$ satisfies Moufang's theorem. This solves a problem posed by Andrew Rajah.

A division sudoku is a latin square whose all six conjugates are sudoku squares. We enumerate division sudokus up to a suitable equivalence, introduce powerful invariants of division sudokus, and also study latin squares that are division sudokus with respect to multiple partitions at the same time. We use nearfields and affine geometry to construct division sudokus of prime power rank that are rich in sudoku partitions.

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

Let $q$ be an odd prime power and suppose that $a,b\in\mathbb{F}_q$ are such that $ab$ and $(1{-}a)(1{-}b)$ are nonzero squares. Let $Q_{a,b} = (\mathbb{F}_q,*)$ be the quasigroup in which the operation is defined by $u*v=u+a(v{-}u)$ if $v-u$ is a square, and $u*v=u+b(v{-}u)$ is $v-u$ is a nonsquare. This quasigroup is called maximally nonassociative if it satisfies $x*(y*z) = (x*y)*z$ $\Leftrightarrow$ $x=y=z$. Denote by $\sigma(q)$ the number of $(a,b)$ for which $Q_{a,b}$ is maximally nonassociative. We show that there exist constants $\alpha \approx 0.02908$ and $\beta \approx 0.01259$ such that if $q\equiv 1 \bmod 4$, then $\lim \sigma(q)/q^2 = \alpha$, and if $q \equiv 3 \bmod 4$, then $\lim \sigma(q)/q^2 = \beta$.

We introduce the concept of propagating equations and focus on the case of associativity propagating in varieties of loops. An equation $\varepsilon$ propagates in an algebra $X$ if $\varepsilon(\overrightarrow y)$ holds whenever $\varepsilon(\overrightarrow x)$ holds and the elements of $\overrightarrow y$ are contained in the subalgebra of $X$ generated by $\overrightarrow x$. If $\varepsilon$ propagates in $X$ then it propagates in all subalgebras and products of $X$ but not necessarily in all homomorphic images of $X$. If $\mathcal V$ is a variety, the propagating core $\mathcal V_{[\varepsilon]} = \{X\in\mathcal V:\varepsilon$ propagates in $X\}$ is a quasivariety but not necessarily a variety. We prove by elementary means Goursat's Lemma for loops and describe all subdirect products of $X^k$ and all finitely generated loops in $\mathbf{HSP}(X)$ for a nonabelian simple loop $X$. If $\mathcal V$ is a variety of loops in which associativity propagates, $X$ is a finite loop in which associativity propagates and every subloop of $X$ is either nonabelian simple or contained in $\mathcal V$, then associativity propagates in $\mathbf{HSP}(X)\lor\mathcal V$. We study the propagating core $\mathcal S_{[x(yz)=(xy)z]}$ of Steiner loops with respect to associativity. While this is not a variety, we exhibit many varieties contained in $\mathcal S_{[x(yz)=(xy)z]}$, each providing a solution to Rajah's problem, i.e., a variety of loops not contained in Moufang loops in which Moufang Theorem holds., Comment: Minor updates to version 1

A quasigroup $Q$ is called maximally nonassociative if for $x,y,z\in Q$ we have that $x\cdot (y\cdot z) = (x\cdot y)\cdot z$ only if $x=y=z$. We show that, with finitely many exceptions, there exists a maximally nonassociative quasigroup of order $n$ whenever $n$ is not of the form $n=2p_1$ or $n=2p_1p_2$ for primes $p_1,p_2$ with $p_1\le p_2<2p_1$.

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 study a generalization of the classical correspondence between homogeneous quadratic polynomials, quadratic forms, and symmetric/alternating bilinear forms to forms in $n$ variables. The main tool is combinatorial polarization, and the approach is applicable even when $n!$ is not invertible in the underlying field.

We enumerate all dissections of an equilateral triangle into smaller equilateral triangles up to size 20, where each triangle has integer side lengths. A perfect dissection has no two triangles of the same side, counting up- and down-oriented triangles as different. We computationally prove W. T. Tutte's conjecture that the smallest perfect dissection has size 15 and we find all perfect dissections up to size 20., Comment: Final version sent to journal.

Let $T = (T^{\textstyle \ast}, T^{\scriptscriptstyle \triangle})$ be a spherical latin bitrade. With each $a=(a_1,a_2,a_3)\in T^{\textstyle \ast}$ associate a set of linear equations $\eq(T,a)$ of the form $b_1+b_2=b_3$, where $b = (b_1,b_2,b_3)$ runs through $T^{\textstyle \ast} \setminus \{a\}$. Assume $a_1 = 0 = a_2$ and $a_3 = 1$. Then $\eq(T,a)$ has in rational numbers a unique solution $b_i = \bar b_i$. Suppose that $\bar b_i \ne \bar c_i$ for all $b,c \in T^{\textstyle \ast}$ such that $b_i \ne c_i$ and $i \in \{1,2,3\}$. We prove that then $T^{\scriptscriptstyle \triangle}$ can be interpreted as a dissection of an equilateral triangle. We also consider group modifications of latin bitrades and show that the methods for generating the dissections can be used for a proof that $T^{\textstyle \ast}$ can be embedded into the operational table of a finite abelian group, for every spherical latin bitrade $T$.

A latin bitrade (T1, T2) is a pair of partial latin squares which are disjoint, occupy the same set of non-empty cells, and whose corresponding rows and columns contain the same set of entries. A genus may be associated to a latin bitrade by constructing an embedding of the underlying graph in an oriented surface. We report computational enumeration results on the number of spherical (genus 0) latin bitrades up to size 24., Comment: Includes source code

Let $Q$ be a Buchsteiner loop. We describe the associator calculus in three variables, and show that $|Q| \ge 32$ if $Q$ is not conjugacy closed. We also show that $|Q| \ge 64$ if there exists $x \in Q$ such that $x^2$ is not in the nucleus of $Q$. Furthermore, we describe a general construction that yields all proper Buchsteiner loops of order 32. Finally, we produce a Buchsteiner loop of order 128 that is nilpotency class 3 and possesses an abelian inner mapping group., Comment: 22 pages

Buchsteiner loops are those which satisfy the identity $x\backslash (xy \cdot z) = (y \cdot zx)/ x$. We show that a Buchsteiner loop modulo its nucleus is an abelian group of exponent four, and construct an example where the factor achieves this exponent., Comment: 33 pages

A latin bitrade is a pair of partial latin squares which are disjoint, occupy the same set of non-empty cells, and whose corresponding rows and columns contain the same set of entries. Dr\'apal (\cite{Dr9}) showed that a latin bitrade is equivalent to three derangements whose product is the identity and whose cycles pairwise have at most one point in common. By letting a group act on itself by right translation, we show how some latin bitrades may be derived from groups without specifying an independent group action. Properties of latin trades such as homogeneousness, minimality (via thinness) and orthogonality may also be encoded succinctly within the group structure. We apply the construction to some well-known groups, constructing previously unknown latin bitrades. In particular, we show the existence of minimal, $k$-homogeneous latin trades for each odd $k\geq 3$. In some cases these are the smallest known such examples., Comment: 23 pages

Two constructions due to Dr\'apal produce a group by modifying exactly one quarter of the Cayley table of another group. We present these constructions in a compact way, and generalize them to Moufang loops, using loop extensions. Both constructions preserve associators, the associator subloop, and the nucleus. We conjecture that two Moufang 2-loops of finite order $n$ with equivalent associator can be connected by a series of constructions similar to ours, and offer empirical evidence that this is so for $n=16$, 24, 32; the only interesting cases with $n\le 32$. We further investigate the way the constructions affect code loops and loops of type $M(G, 2)$. The paper closes with several conjectures and research questions concerning the distance of Moufang loops, classification of small Moufang loops, and generalizations of the two constructions., Comment: 20 pages