On loops and groups
by
Piroska Csörgö
Eötvös University, Budapest, Hungary

A finite groupoid Q is called loop, if it contains a neutral element and there is unique division in it. The multiplication group M(Q) of a loop Q is the group generated by the right and left translations. The stabiliser of the neutral element is I(Q).

T. Kepka and M. Niemenmaa proved the following:

THEOREM. A group G is isomorphic to the multiplication group of a loop iff there exists a subgroup H satisfying core H=1 and there are H-connected transversals A and B (t.i. H contains [A, B]), furthermore G=áA, Bñ.

In 1996 A. Vesanen showed that the solvability of M(Q) implies the solvability of Q.

The problem is the following: what property of I(Q) garantees the solvability of M(Q). Using Kepka's and Niemenmaa's theorem, this question can be converted to a group theroretical problem. If I(Q) is abelian or a dihedral 2-group, the answer is yes. The result is known in some special cases when the order of I(Q) is the product of two primes. Our new result with M. Niemenmaa is:

Theorem. M(Q) is solvable in the following cases:

a)

if the order of I(Q) is 2p with p an odd prime;

b)

if I(Q) is a dihedral group of order 2pⁿ with p a prime.

Date received: June 30, 1999