Related papers: When is the commutant of a Bol loop a subloop?

We study the probability of a given element, in the commutator subgroup of a group, to be equal to a commutator of two randomly chosen group elements, and compute explicit formulas for calculating this probability for some interesting…

For G a group and g in G, we define mappings pg(G) and lg(G) from G into G by pg(x)=[x,g] and lg(x)=[g,x]. We let P(G) and L(G) denote the subsemigroups of the set of all mappings from G to G generated by {pg: g in G} and {lg: g in G},…

A subsemigroup S of a semigroup Q is a left order in Q and Q is a semigroup of left quotients of S if every element of Q can be expressed as a# b where a and b are elements of S and if, in addition, every element of S that is square…

For most (and possibly all) non-associative finite simple Moufang loops, three generators of order 3 can be chosen so that each two of them generate a group isomorphic to $(3, 3 | 3, p)$. The subgroup structure of $(3, 3 | 3, p)$ depends on…

Let $B$ be a finitely generated algebra over a field $k$. Then $B$ is called a Jacobson algebra if every semiprime ideal of $B$ is semiprimitive. We will discuss several conditions, all involving the commutant of simple $B$-modules, which…

Although little can be gleaned about a loop with the property that its squares are, say, left nuclear ($xx\cdot yz = (xx\cdot y)z$), if its squares are also, say, middle nuclear ($(x\cdot yy)z = x(yy\cdot z)$), then the loop exhibits more…

Interpolation theory for complex polynomials is well understood. In the non-commutative quaternionic setting, the polynomials can be evaluated "on the left" and "on the right". If the interpolation problem involves interpolation conditions…

While Jordan algebras are commutative, their non-associativity makes it so that the Jordan product operators do not necessarily commute. When the product operators of two elements commute, the elements are said to operator commute. In some…

Non-associative finite invertible loops (NAFIL) are loops whose every element has a unique two-sided inverse. Not much is known about the class of NAFIL loops which includes the familiar IP (Inverse Property), Moufang, and Bol loops. Our…

Let $k$ be an algebraically closed field of characteristic $p>0$. For a loop $\circlearrowleft$, denote its path coalgebra by $k\circlearrowleft$. In this paper, all the finite-dimensional commutative Hopf algebras over the sub coalgebras…

We study so called regular Lie algebras, i.e. Lie algebras in which each nonzero element is regular. We make a connection with an open problem whether any element of reduced trace zero in a simple associative algebra is a commutator.

Commutative totally ordered monoids abound, number systems for example. When the monoid is not assumed commutative, one may be hard pressed to find an example. One suggested by Professor Orr Shalit are the countable ordinals with addition.…

A mixed lattice is a partially ordered set with two mixed partial orderings that are linked by asymmetric upper and lower envelopes. These notions generalize the join and meet operations of a lattice. In the present paper, we study…

Let $R$ be a ring and let $n\ge 2$. We discuss the question of whether every element in the matrix ring $M_n(R)$ is a product of (additive) commutators $[x,y]=xy-yx$, for $x,y\in M_n(R)$. An example showing that this does not always hold,…

Let p be a prime. Every finite group G has a normal series each of whose quotients either is p-soluble or is a direct product of nonabelian simple groups of orders divisible by p. The non-p-soluble length of G is defined as the minimal…

A finite non-abelian group $G$ is called commuting integral if the commuting graph of $G$ is integral. In this paper, we show that a finite group is commuting integral if its central factor is isomorphic to ${\mathbb{Z}}_p \times…

Given a group $G$ and elements $x_1,x_2,\dots, x_\ell\in G$, the commutator of the form $[x_1,x_2,\dots, x_\ell]$ is called a commutator of length $\ell$. The present paper deals with groups having only finitely many commutators of length…

Let $G$ be a finite group. A coprime commutator in $G$ is any element that can be written as a commutator $[x,y]$ for suitable $x,y\in G$ such that $\pi(x)\cap\pi(y)=\emptyset$. Here $\pi(g)$ denotes the set of prime divisors of the order…

Cumulants linearize convolution of measures. We use a formula of Good to define noncommutative cumulants in a very general setting.It turns out that the essential property needed is exchangeability of random variables. Roughly speaking the…

Let G be a group. A subset X of G is a set of pairwise non-commuting elements if xy is not equal to yx for any two distinct elements x and y in X. If |X|>=|Y| for any other set of pairwise non-commuting elements Y in G, then X is said to be…