Goursats Theorem and the Zassenhaus Lemma

Abstract

In this paper we study generalized homomorphisms between two algebras, namely the binary relations whose graphs are subalgebras of the direct product of the given algebras. In 1897 Goursat proved that every subgroup of the direct product of two groups is determined by an isomorphism between factor groups of subgroups of the given groups (10, §§11, 12; 25, pp. 15, 16). A like result is here shown for a general class of algebras, including loops and quasigroups, by a method due to Riguet (22). This result is used to obtain general forms of the Zassenhaus lemma and the Jordan- Hälder-Schreier theorem for normal series (26, §9).