A loop $Q$ is said to be left conjugacy closed if the set $A=\{L_x/x\in Q\}$ is closed under conjugation. Let $Q$ be an LCC loop, let $ℒ$ and $ℛ$ be the left and right multiplication groups of $Q$ respectively, and let $I\left(Q\right)$ be its inner mapping group, $M\left(Q\right)$ its multiplication group. By Drápal’s theorem [3, Theorem 2.8] there exists a homomorphism $\Lambda :ℒ\to I\left(Q\right)$ determined by $L_x\to R_x^{-1}{L}_x$. In this short note we examine different possible extensions of this $\Lambda$ and the uniqueness of these extensions.

It is well-known that a based space is of the weak homotopy type of a loop space iff it is a grouplike algebra over an $A_{\infty }$-operad. The classical model for such an operad consists of Stasheff’s associahedra. The present paper describes a similar recognition principle for free loop spaces. Let $𝒫$ be an operad, $M$ a $𝒫$-module and $U$ a $𝒫$-algebra. An $M$-trace over $U$ consists of a space $V$ and a module homomorphism $T:M\to {\text{End}}_{U,V}$ over the operad homomorphism $𝒫\to {\text{End}}_U$ given by the algebra structure on $U$. Let ${𝒞}_1$ be the...

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 $i$, $j$, $k\in \{-1,1\}$. 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 $M(G,2)$.

Let $Q$ be a finite commutative loop and let the inner mapping group $I\left(Q\right)\cong C_{p^n}\times C_{p^n}$, where $p$ is an odd prime number and $n\ge 1$. We show that $Q$ is centrally nilpotent of class two.