##
**Some results in the theory of quasigroups.**
*(English)*
Zbl 0063.00635

Introduction: The concept of isotopy, recently introduced by A. A. Albert in connection with the theory of linear non-associative algebras, appears to have its value in the theory of quasigroups. Conversely, the author has been able to use quasigroups in the study of linear non-associative algebras. The present paper is primarily intended as an illustration of the usefulness of isotopy in quasigroup-theory and as groundwork for a later paper on algebras, but is bounded by neither of these aspects.

The first two sections are devoted to the basic definitions of quasigroup and isotopy, along with some elementary remarks and two fundamental theorems due to Albert. Then there is initiated a study of special types of quasigroup, beginning with quasigroups with the inverse property (I.P. quasigroups). A system \(Q\) of elements \(a,b,\dots\) is called an I.P. quasigroup if it possesses a single-valued binary operation \(ab\) and there exist two one-to-one reversible mappings \(L\) and \(R\) of \(Q\) on itself such that \[ a^L(ab)=(ba)a^R=b\tag{1} \] for every pair of elements \(a,b\) of \(Q\).

Sections 3 to 9 inclusive are devoted in the main to I.P. quasigroups; in particular, to methods of constructing them and to their isotopy properties. In §3 will be found some elementary consequences of the definition (Lemma 1) and a construction of all I.P. quasigroups which are isotopic to a group (Theorem 3).

Lemma 2 (§4) gives necessary and sufficient conditions in order that an I.P. quasigroup may be constructed from a non-associative ring by use of the multiplication \[ x\circ y= x+y+xy.\tag{2} \] In §5 the same matter is pursued further (in the special case that a certain mapping \(J\) of the ring is linear) and leads to a specific construction. As a result we find (Theorem 4) that for every odd prime \(p\) and for every integer \(n\) not less than 7 there exists an I.P. quasigroup \(Q\) of order \(p^n\), which may be regarded as a (non-associative, noncommutative) extension of an Abelian group of order \(p^{n-3}\) by another Abelian group of order \(p^3\). Moreover \(Q\) is a Moufang quasigroup. (This last statement we shall explain in a later paragraph.) After demonstrating that a theory of coset expansions is impossible for general I.P. quasigroups, we show in §6 that for those I.P. quasigroups which are defined as above we may define normal sub-quasigroups in terms of invariant subrings and obtain the usual desirable features of cosets and quotient quasigroups (Theorem 5). Theorem 6, which was suggested by ideas of coset expansion but is otherwise unconnected with the rest of §6, gives an explicit construction of an I.P. quasigroup of index 2 over any I.P. quasigroup with a unique unit element. As a corollary we derive the existence of nonassociative I.P. quasigroups of order \(2^n\) for every integer \(n\) not less than 4.

Section 7 introduces three new types of quasigroup: idempotent quasigroups (those in which every element is idempotent), unipotent quasigroups (quasigroups with unique unit elements in which the square of every element is the unit element), and totally symmetric or T.S. quasigroups (in which a valid equation \(ab=c\) remains true under every permutation of the letters \(a,b,c\)). Quasigroups of the first two types possess the inverse property if and only if they are totally symmetric. The main feature of this section is a construction by which there may be obtained from an idempotent quasigroup of order \(n\) a unipotent quasigroup of order \(n+1\) and conversely. The construction preserves the inverse property. It is also shown that there exist idempotent (unipotent) quasigroups of every finite order except order two (order three).

In §8 the above-mentioned construction is combined with the direct product to obtain a variety of T.S. quasigroups; these of course have the inverse property. In the same section necessary and sufficient conditions are given in order that an I.P. quasigroup with a unique unit element should possess an isotope which is totally symmetric (Theorem 7). It follows that unipotent T.S. quasigroups are isotopic if and only if they are isomorphic (Corollary 3). (The interest of this last result rests in the fact that it is an analogue of a theorem of Albert on groups (Theorem 2).) Theorem 8 gives necessary and sufficient conditions that an idempotent T.S. quasigroup should possess an I.P. isotope with a unit element, and a corollary exhibits an idempotent T.S. quasigroup of order \(2^n-1\) (for every integer \(n\) not less than 3) which does not possess an I.P. isotope with a unit element.

In §9 further attention is given to the special I.P. quasigroups, mentioned above, which D. C. Murdoch has called Moufang quasigroups after their originator, Miss R. Moufang. A Moufang quasigroup is a quasigroup with a unique unit element in which the elements obey the mild associative law \[ a(b\cdot cb)= (ab\cdot c)b.\tag{3} \] The main result of the section (Theorem 9) is to the following effect. A necessary and sufficient condition that every isotope, which possesses a unique unit element, of a quasigroup \(Q\) with a unit element should have the inverse property is that \(Q\) be a Moufang quasigroup. In particular, an isotope of a Moufang quasigroup is a Moufang quasigroup provided it has a unit element. A similar result (Theorem 10) holds for alternative fields. The section also contains a brief sketch of the previously known theory of Moufang quasigroups.

Section 10 is devoted to the so-called Abelian quasigroups of D. C. Murdoch. First it is shown that every Abelian quasigroup is isotopic to an Abelian group, unique in the sense of isomorphism (Theorem 11). This result is essentially Murdoch’s. Next, an explicit construction is given of every isotope of an Abelian group which is an Abelian quasigroup in the sense of Murdoch (Theorem 12).

Section 11, which was prompted by an erroneous remark in a paper of Murdoch’s, contains necessary and sufficient conditions that the direct product of two finite quasigroups should possess no sub-quasigroup except itself. The statement of these conditions (Theorem 13) is given in terms of the invariant complexes of G. N. Garrison.

The first two sections are devoted to the basic definitions of quasigroup and isotopy, along with some elementary remarks and two fundamental theorems due to Albert. Then there is initiated a study of special types of quasigroup, beginning with quasigroups with the inverse property (I.P. quasigroups). A system \(Q\) of elements \(a,b,\dots\) is called an I.P. quasigroup if it possesses a single-valued binary operation \(ab\) and there exist two one-to-one reversible mappings \(L\) and \(R\) of \(Q\) on itself such that \[ a^L(ab)=(ba)a^R=b\tag{1} \] for every pair of elements \(a,b\) of \(Q\).

Sections 3 to 9 inclusive are devoted in the main to I.P. quasigroups; in particular, to methods of constructing them and to their isotopy properties. In §3 will be found some elementary consequences of the definition (Lemma 1) and a construction of all I.P. quasigroups which are isotopic to a group (Theorem 3).

Lemma 2 (§4) gives necessary and sufficient conditions in order that an I.P. quasigroup may be constructed from a non-associative ring by use of the multiplication \[ x\circ y= x+y+xy.\tag{2} \] In §5 the same matter is pursued further (in the special case that a certain mapping \(J\) of the ring is linear) and leads to a specific construction. As a result we find (Theorem 4) that for every odd prime \(p\) and for every integer \(n\) not less than 7 there exists an I.P. quasigroup \(Q\) of order \(p^n\), which may be regarded as a (non-associative, noncommutative) extension of an Abelian group of order \(p^{n-3}\) by another Abelian group of order \(p^3\). Moreover \(Q\) is a Moufang quasigroup. (This last statement we shall explain in a later paragraph.) After demonstrating that a theory of coset expansions is impossible for general I.P. quasigroups, we show in §6 that for those I.P. quasigroups which are defined as above we may define normal sub-quasigroups in terms of invariant subrings and obtain the usual desirable features of cosets and quotient quasigroups (Theorem 5). Theorem 6, which was suggested by ideas of coset expansion but is otherwise unconnected with the rest of §6, gives an explicit construction of an I.P. quasigroup of index 2 over any I.P. quasigroup with a unique unit element. As a corollary we derive the existence of nonassociative I.P. quasigroups of order \(2^n\) for every integer \(n\) not less than 4.

Section 7 introduces three new types of quasigroup: idempotent quasigroups (those in which every element is idempotent), unipotent quasigroups (quasigroups with unique unit elements in which the square of every element is the unit element), and totally symmetric or T.S. quasigroups (in which a valid equation \(ab=c\) remains true under every permutation of the letters \(a,b,c\)). Quasigroups of the first two types possess the inverse property if and only if they are totally symmetric. The main feature of this section is a construction by which there may be obtained from an idempotent quasigroup of order \(n\) a unipotent quasigroup of order \(n+1\) and conversely. The construction preserves the inverse property. It is also shown that there exist idempotent (unipotent) quasigroups of every finite order except order two (order three).

In §8 the above-mentioned construction is combined with the direct product to obtain a variety of T.S. quasigroups; these of course have the inverse property. In the same section necessary and sufficient conditions are given in order that an I.P. quasigroup with a unique unit element should possess an isotope which is totally symmetric (Theorem 7). It follows that unipotent T.S. quasigroups are isotopic if and only if they are isomorphic (Corollary 3). (The interest of this last result rests in the fact that it is an analogue of a theorem of Albert on groups (Theorem 2).) Theorem 8 gives necessary and sufficient conditions that an idempotent T.S. quasigroup should possess an I.P. isotope with a unit element, and a corollary exhibits an idempotent T.S. quasigroup of order \(2^n-1\) (for every integer \(n\) not less than 3) which does not possess an I.P. isotope with a unit element.

In §9 further attention is given to the special I.P. quasigroups, mentioned above, which D. C. Murdoch has called Moufang quasigroups after their originator, Miss R. Moufang. A Moufang quasigroup is a quasigroup with a unique unit element in which the elements obey the mild associative law \[ a(b\cdot cb)= (ab\cdot c)b.\tag{3} \] The main result of the section (Theorem 9) is to the following effect. A necessary and sufficient condition that every isotope, which possesses a unique unit element, of a quasigroup \(Q\) with a unit element should have the inverse property is that \(Q\) be a Moufang quasigroup. In particular, an isotope of a Moufang quasigroup is a Moufang quasigroup provided it has a unit element. A similar result (Theorem 10) holds for alternative fields. The section also contains a brief sketch of the previously known theory of Moufang quasigroups.

Section 10 is devoted to the so-called Abelian quasigroups of D. C. Murdoch. First it is shown that every Abelian quasigroup is isotopic to an Abelian group, unique in the sense of isomorphism (Theorem 11). This result is essentially Murdoch’s. Next, an explicit construction is given of every isotope of an Abelian group which is an Abelian quasigroup in the sense of Murdoch (Theorem 12).

Section 11, which was prompted by an erroneous remark in a paper of Murdoch’s, contains necessary and sufficient conditions that the direct product of two finite quasigroups should possess no sub-quasigroup except itself. The statement of these conditions (Theorem 13) is given in terms of the invariant complexes of G. N. Garrison.

### MSC:

20N05 | Loops, quasigroups |