In this paper we consider finite loops whose inner mapping groups are nilpotent. We first consider the case where the inner mapping group of a loop is the direct product of a dihedral group of order and an abelian group. Our second result deals with the case where is a -loop and is a nilpotent group whose nonabelian Sylow subgroups satisfy a special condition. In both cases it turns out that is centrally nilpotent.
Using a lemma on subnormal subgroups, the problem of nilpotency of multiplication groups and inner permutation groups of centrally nilpotent loops is discussed.
Using a construction of commutative loops with metacyclic inner mapping group and trivial center described by A. Drápal, we enumerate presumably all such loops of order , for and primes.
Some results concerning congruence relations on partially ordered quasigroups (especially, Riesz quasigroups) and ideals of partially ordered loops are presented. These results generalize the assertions which were proved by Fuchs in [5] for partially ordered groups and Riesz groups.
Let be a finite group with a dicyclic subgroup . We show that if there exist -connected transversals in , then is a solvable group. We apply this result to loop theory and show that if the inner mapping group of a finite loop is dicyclic, then is a solvable loop. We also discuss a more general solvability criterion in the case where is a certain type of a direct product.
We show that finite commutative inverse property loops may not have nonabelian dihedral 2-groups as their inner mapping group.
It is an open question whether the variety generated by the left divisible left distributive groupoids coincides with the variety generated by the left distributive left quasigroups. In this paper we prove that every left divisible left distributive groupoid with the mapping surjective lies in the variety generated by the left distributive left quasigroups.
We show that finite commutative inverse property loops with elementary abelian inner mapping groups of order are centrally nilpotent of class at most two.
We show that finite commutative inverse property loops with elementary abelian inner mapping groups of order are centrally nilpotent of class at most two.
Let be a finite commutative loop and let the inner mapping group , where is an odd prime number and . We show that is centrally nilpotent of class two.
In this paper we consider finite loops and discuss the following problem: Which groups are (are not) isomorphic to inner mapping groups of loops? We recall some known results on this problem and as a new result we show that direct products of dihedral 2-groups and nontrivial cyclic groups of odd order are not isomorphic to inner mapping groups of finite loops.
We investigate the situation that the inner mapping group of a loop is of order which is a product of two small prime numbers and we show that then the loop is soluble.
We give sufficient and in some cases necessary conditions for the conjugacy closedness of provided the commutativity of . We show that if for some loop , and are abelian groups, then is a CC loop, consequently has nilpotency class at most three. We give additionally some reasonable conditions which imply the nilpotency of the multiplication group of class at most three. We describe the structure of Buchsteiner loops with abelian inner mapping groups.
Multiplication groups of (finite) loops with commuting inner permutations are investigated. Special attention is paid to the normal closure of the abelian permutation group.