On finite loops whose inner mapping groups are Abelian

Abstract

ON FINITE LOOPS WHOSE INNER MAPPING GROUPS ARE ABELIANMARKKU NIEMENMAA Loops are nonassociative algebras which can be investigated by using their multiplication groups and inner mapping groups. If the inner mapping group of a loop is finite and Abelian, then the multiplication group is a solvable group. It is clear that not all finite Abelian groups can occur as inner mapping groups of loops. In this paper we show that certain finite Abelian groups with a special structure are not isomorphic to inner mapping groups of finite loops. We use our results and show how to construct solvable groups which are not isomorphic to multiplication groups of loops.

Small loops of nilpotency class 3 with commutative inner mapping groups

“…If Q is a p-loop with cl(Q) > 2 then Inn Q is isomorphic neither to Z p × Z p , by [4,Theorem 4.2], nor to Z p × Z p × Z p , by [2]. If Q is finite, k ≥ 2, p is an odd prime [9] or an even prime [4,Theorem 4.1], then Inn Q is not isomorphic to Z p k × Z p . On the positive side, taking the examples of [5] and of this paper into account, we now know that Inn Q can be isomorphic to (Z 2 ) 6 or to (Z 4 ) 2 × (Z 2 ) 2 when Q is a 2-loop with cl(Q) > 2.…”

Abelian groups as inner mapping groups of loops

“…Groups G with this structure arise naturally as the multiplication groups of loops, with H the inner mapping group of the loop, and have been studied extensively by Niemenmaa and others. We refer the interested reader to [5,6,7,8] in particular for the translation of our results to loops. The objects we study in this paper are 4-tuples (G,H,A,B), where G is a group, H a subgroup of G and A, B transversals for H in G. Niemenmaa and his collaborators have been interested in particular in groups satisfying the following hypothesis:…”

A nuclear construction of loops with small inner mapping groups

“…Thus q divides p -1 and Inn Q is not commutative if it is of order pq. The solvability of Mit Q for such loops was studied extensively by NIEMENMAA et al in a series of papers [8,9,1,7]. It was proved in [2] that Mit Q has to be always solvable, and that the structure of Q has to fulfil a number of constraints.…”