Multiplication groups of commutative automorphic \(p\)-loops of odd order are \(p\)-groups.
The study of automorphic loops (A-loops) was initiated by \textit{R. H. Bruck} and \textit{L. J. Paige}, [Ann. Math. (2) 63, 308-323 (1956; Zbl 0074.01701)]. In this paper the author proves without using the nilpotence that the multiplication group of a finite commutative A-loop of order a power of an odd prime \(p\) is a \(p\)-group. Consequently the loop has nontrivial center which implies the central nilpotence of the loop. The author proves this result by applying the theory of connected transversals that was introduced by \textit{M. Niemenmaa} and \textit{T. Kepka} [in J. Algebra 135, No. 1, 112-122 (1990; Zbl 0706.20046)].
scientific article; zbMATH DE number 6037589