The (total) multiplication group of a quasigroup Q is the group generated by all (left, right and middle) left and right translations of Q. Natural transformations of quasigroups (loops) are parastrophy (inversion), isotopy and isostrophy (the product of a parastrophy and an isotopy)(see [1-2]). It is known that multiplication (total multiplication) groups of isotopic (isostrophic) loops are isomorphic (see [3]). Invariant properties under the isotopy of loops are called universal properties. (Left, right) middle Bol loops are loops with universal (left inverse, right inverse) anti-automorphic inverse property. Moreover, middle Bol loops are isostrophic to the left and to the right Bol loops. Connections between multiplication and total multiplication groups of loops (in particular, of left, right and middle Bol loops) are considered in the present talk. A characterization of the commutative middle Bol loops is given. This is a joint work with A. Drapal.