Middle Bruck Loops and the Total Multiplication Group

Abstract

Let Q be a loop. The mappings $x\mapsto ax$, $x\mapsto xa$ and $x\mapsto a/x$ are denoted by $L_a$, $R_a$ and $D_a$, respectively. The loop is said to be middle Bruck if for all $a,b \in Q$ there exists $c\in Q$ such that $D_aD_bD_a = D_c$. The right inverse of Q is the loop with operation $x/(y\backslash 1)$. It is proved that Q is middle Bruck if and only if the right inverse of Q is left Bruck (i.e., a left Bol loop in which $(xy)^{-1}= x^{-1}y^{-1}$). Middle Bruck loops are characterized in group theoretic language as transversals T to $H\le G$ such that $\langle T \rangle = G$, $T^G =T$ and $t^2=1$ for each $t\in T$. Other results include the fact that if Q is a finite loop, then the total multiplication group $\langle L_a,R_a,D_a; a\in Q\rangle$ is nilpotent if and only if Q is a centrally nilpotent 2-loop, and the fact that total multiplication groups of paratopic loops are isomorphic.