Hamiltonian Loops

A Hamilton loop is a loop in which every subloop is normal. This definition is somewhat broader than the analogous definition of group theory where a Hamiltonian group is restricted to be non-Abelian. The structure of a Hamiltonian group is well known [4, pp. 129-131 ]l —it is the direct product of an Abelian group whose elements have odd order, an Abelian group with exponent 2, and a quaternion group. The purpose of this paper is to investigate the structure of Hamiltonian loops. It is relatively easy to construct loops of any order (except 4) which have no proper subloops and which are therefore trivially Hamiltonian. With a more careful construction it may be shown that given any series of integers Mi, re2, • • • , »t where re.^4, i=\, 2, • • • , k, there exists a loop Nk of order rei«2 • • • nk with subloops Ni, N2, • • • , Nk-i of order (wi), (rei«2), ■••,(«!••• «¡t-i) respectively such that NiEN2E • • • ENk, each loop Nt, i<k, is normal in Nk, and A^ contains no other proper subloops.2 Thus we can construct a Hamiltonian loop with a prescribed series of composition. It becomes apparent that for a complete structure theory some additional hypotheses are necessary. In Theorem I a necessary and sufficient condition that the direct product of Hamiltonian loops be Hamiltonian is obtained. In Theorems II to VII the structure of a Hamiltonian loop under successively greater restrictions is discussed. In each case where an additional restriction is added to the loop, an example may be constructed to show that there exist loops satisfying all previous conditions but not the additional one, and which do not satisfy the resulting theorem. The simplest assumption yielding significant results is that the loop be power associative. We shall list a few familiar facts concerning normal subloops. The following lemma is due to Brück [2, p. 256].