Related papers: A class of Bol loops with a subgroup of index two

Let $G$ be a finite group and $C_2$ the cyclic group of order 2. Consider the 8 multiplicative operations $(x,y)\mapsto (x^iy^j)^k$, where $i$, $j$, $k\in\{-1, 1\}$. Define a new multiplication on $G\times C_2$ by assigning one of the above…

A quasigroup identity is of Bol-Moufang type if two of its three variables occur once on each side, the third variable occurs twice on each side, the order in which the variables appear on both sides is the same, and the only binary…

The existence of finite simple non-Moufang Bol loops was considered as one of the main open problems in the theory of loops and quasigroups. In this paper, we present a class of proper simple Bol loops. This class also contains finite and…

In this paper we give an infinite class of finite simple right Bol loops of exponent 2. The right multiplication group of these loops is an extension of an elementary Abelian 2-group by $S_5$. The construction uses the description of the…

We partially answer two questions of Goodaire by showing that in a finite, strongly right alternative ring, the set of units (if the ring is with unity) is a Bol loop under ring multiplication, and the set of quasiregular elements is a Bol…

We describe all constructions for loops of Bol-Moufang type analogous to the Chein construction $M(G,*,g_0)$ for Moufang loops.

We initiate (co)homology theory for quasigroups of Bol-Moufang type based on analysis of their extensions by affine quasigroups of the same type. We use these extensions to define second and third boundary operations, $\partial_2(x,y)$ and…

A Bol loop is a loop that satisfies the Bol identity $(xy.z)y=x(yz.y)$. If $L$ is a loop and $f:L\to L$ is a bijection such that $f(xy)\in\{f(x)f(y),f(y)f(x)\}$, for every $x$, $y\in L$, then $f$ is called a half-automorphism of $L$. In…

Suppose $C(G)$ denotes the set of all cyclic subgroups of a finite group $G$, and $\mathcal{O}_{2}(G)$ denotes the number of elements of order $2$ in $G$. In [Marius T., Finite groups with a certain number of cyclic subgroups. The American…

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…

Let $p>q$ be odd primes. We classify Bol loops and Bruck loops of order $pq$ up to isotopism. When $q$ does not divide $p^2-1$, the only Bol loop (and hence the only Bruck loop) of order $pq$ is the cyclic group of order $pq$. When $q$…

If the left multiplication group of a loop is simple, then the loop is simple. We use this observation to give examples of infinite simple Bol loops.

We investigate the multiplication group of a special class of quasigroup called AG-group. We prove some interesting results such as: the multiplication group of an AG-group of order n is non-abelian group of order 2n and its left section is…

In this paper we investigate the Bol loops and connected with them groups. We prove an analog of the Doro's theorem for Moufang loops and find a criterion for simplicity of Bol loops. One of the main results obtained is the following: if…

A real form $G_0$ of a complex semisimple Lie group $G$ has only finitely many orbits in any given compact $G$-homogeneous projective algebraic manifold $Z=G/Q$. A maximal compact subgroup $K_0$ of $G_0$ has special orbits $C$ which are…

A loop $(X,\circ)$ is said to be a Bruck loop if it satisfies the (right) Bol identity $((z\circ x)\circ y)\circ x = z\circ ((x\circ y)\circ x)$ and the automorphic inverse property $(x\circ y)^{-1}=x^{-1}\circ y^{-1}$. If $X$ is a finite…

A loop $(Q,\cdot,\backslash,/)$ is called a middle Bol loop if it obeys the identity $x(yz\backslash x)=(x/z)(y\backslash x)$. In this paper, some new algebraic properties of a middle Bol loop are established. Four bi-variate mappings…

We give here the exact maximal subgroup growth of two classes of polycyclic groups. Let $G_k = \langle x_1, x_2, ..., x_k \mid x_ix_jx_i^{-1}x_j \text{ for all } i < j \rangle$. So $G_k = \mathbb{Z} \rtimes (\mathbb{Z} \rtimes (\mathbb{Z}…

Using the relations between the theory of differentiable Bol loops and the theory of affine symmetric spaces we classify all connected differentiable Bol loops having an at most $9$-dimensional semi-simple Lie group as the group…

Let $M$ be a closed, connected, orientable topological four-manifold with $H_1(M)$ nontrivial and free abelian, $b_2(M)\ne 0, 2$, and $\chi(M)\ne 0$. We show that if $G$ is a finite group of 2-rank $\le 1$ which admits a homologically…