Drapal, AlesSyrbu, Parascovia2023-10-042023-10-042022DRAPAL, Ales, SYRBU, Parascovia. Middle Bruck Loops and the Total Multiplication Group. In: Results in Mathematics, 2022, nr. 4(77), p. 0. ISSN 1422-6383. DOI: 10.1007/s00025-022-01716-21422-6383https://doi.org/10.1007/s00025-022-01716-2https://msuir.usm.md/handle/123456789/11112Let Q be a loop. The mappings x↦ ax, x↦ xa and x↦ a/ x are denoted by La, Ra and Da, respectively. The loop is said to be middle Bruck if for all a, b∈ Q there exists c∈ Q such that DaDbDa= Dc. The right inverse of Q is the loop with operation x/ (y\ 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- 1y- 1). Middle Bruck loops are characterized in group theoretic language as transversals T to H≤ G such that ⟨ T⟩ = G, TG= T and t2= 1 for each t∈ T. Other results include the fact that if Q is a finite loop, then the total multiplication group⟨ La, Ra, Da; a∈ Q⟩ 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.enBruck loopmiddle Bol loopparatopytotal multiplication groupArticle