Markku Niemenmaa

On loops and their multiplication groups

We say that a groupoid $Q$ is a loop if $Q$ has a unique division and a neutral element (thus loops are nonassociative versions of groups). The mappings $L_a(x)=ax$ and $R_a(x)=xa$ define two permutations on $Q$ for every $a \in Q$ and the permutation group $M(Q)= \langle L_a , R_a : a \in Q \rangle$ is called the multiplication group of $Q$. By $I(Q)$ we denote the stabilizer of the neutral element and $I(Q)$ is called the inner mapping group of $Q$. Many properties of loops can be reduced to the properties of connected transversals in the multiplication group. The precise definition is as follows: If $G$ is a group, $H \le G$ and $A$ and $B$ are two left transversals to $H$ in $G$ such that the commutator subgroup $[A,B]$ is contained in $H$ then we say that $A$ and $B$ are $H$-connected in $G$. By $H_G$ we denote the core of $H$ in $G$ (the largest normal subgroup of $G$ contained in $H$). If $Q$ is a loop then it is rather easy to see that the core of $I(Q)$ in $M(Q)$ is trivial and $A= \{L_a: a \in Q \}$ and $B= \{R_a: a \in Q\}$ are $I(Q)$-connected transversals in $M(Q)$. In my talk I shall explore the relation between loops, their multiplication groups and connected transversals.