### $(*)$-groups and pseudo-bad groups.

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Let $G$ be a finite group and ${C}_{2}$ the cyclic group of order $2$. Consider the $8$ multiplicative operations $(x,y)\mapsto {\left({x}^{i}{y}^{j}\right)}^{k}$, where $i$, $j$, $k\in \{-1,\phantom{\rule{0.166667em}{0ex}}1\}$. Define a new multiplication on $G\times {C}_{2}$ by assigning one of the above $8$ multiplications to each quarter $(G\times \{i\left\}\right)\times (G\times \{j\left\}\right)$, for $i,j\in {C}_{2}$. We describe all situations in which the resulting quasigroup is a Bol loop. This paper also corrects an error in P. Vojtěchovsk’y: On the uniqueness of loops $M(G,2)$.

The purpose of this paper is to prove the existence of a free subgroup of the group of all affine transformations on the plane with determinant 1 such that the action of the subgroup is locally commutative.

We generalize the theory of generic subsets of definably compact definable groups to arbitrary o-minimal structures. This theory is a crucial part of the solution to Pillay's conjecture connecting definably compact definable groups with Lie groups.

Groups are usually axiomatized as algebras with an associative binary operation, a two-sided neutral element, and with two-sided inverses. We show in this note that the same simplicity of axioms can be achieved for some of the most important varieties of loops. In particular, we investigate loops of Bol-Moufang type in the underlying variety of magmas with two-sided inverses, and obtain ``group-like'' equational bases for Moufang, Bol and C-loops. We also discuss the case when the inverses are only...

We show that for no infinite group $G$ the class of abelian-by-$G$ groups is elementary, but, at least when $G$ is an infinite elementary abelian $p$-group (with $p$ prime), the class of groups admitting a normal abelian subgroup whose quotient group is elementarily equivalent to $G$ is elementary.

A long-standing conjecture of Podewski states that every minimal field is algebraically closed. Known in positive characteristic, it remains wide open in characteristic zero. We reduce Podewski's conjecture to the (partially) ordered case, and we conjecture that such fields do not exist. We prove the conjecture in case the incomparability relation is transitive (the almost linear case). We also study minimal groups with a (partial) order, and give a complete classification of...

Quasigroups were originally described combinatorially, in terms of existence and uniqueness conditions on the solutions to certain equations. Evans introduced a universal-algebraic characterization, as algebras with three binary operations satisfying four identities. Now, quasigroups are redefined as heterogeneous algebras, satisfying just two conditions respectively known as hypercommutativity and hypercancellativity.

We give new equations that axiomatize the variety of trimedial quasigroups. We also improve a standard characterization by showing that right semimedial, left F-quasigroups are trimedial.

We describe first-order logic elementary embeddings in a torsion-free hyperbolic group in terms of Sela’s hyperbolic towers. Thus, if $H$ embeds elementarily in a torsion free hyperbolic group $\Gamma $, we show that the group $\Gamma $ can be obtained by successive amalgamations of groups of surfaces with boundary to a free product of $H$ with some free group and groups of closed surfaces. This gives as a corollary that an elementary subgroup of a finitely generated free group is a free factor. We also consider the...