Gyrogroups and left gyrogroups as transversals of a special kind

In this article we study gyrogroups and left gyrogroups as transversals in some suitable groups to its subgroups. These objects were introduced into consideration in a connection with an investigation of analogies between symmetries in the classical mechanics and in the relativistic one. The autho...

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Дата:	2003
Автор:	Kuznetsov, E.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2003
Назва видання:	Algebra and Discrete Mathematics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/155728
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Цитувати:	Gyrogroups and left gyrogroups as transversals of a special kind / E. Kuznetsov // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 3. — С. 53–81. — Бібліогр.: 15 назв. — англ.

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spelling	irk-123456789-1557282019-06-19T01:27:05Z Gyrogroups and left gyrogroups as transversals of a special kind Kuznetsov, E. In this article we study gyrogroups and left gyrogroups as transversals in some suitable groups to its subgroups. These objects were introduced into consideration in a connection with an investigation of analogies between symmetries in the classical mechanics and in the relativistic one. The author introduce some new notions into consideration (for example, a weak gyrotransversal) and give a full description of left gyrogroups (and gyrogroups) in terms of transversal identities. Also he generalize a construction of a diagonal transversal and obtain a set of new examples of left gyrogroups. 2003 Article Gyrogroups and left gyrogroups as transversals of a special kind / E. Kuznetsov // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 3. — С. 53–81. — Бібліогр.: 15 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 20N05, 20N15. http://dspace.nbuv.gov.ua/handle/123456789/155728 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	In this article we study gyrogroups and left gyrogroups as transversals in some suitable groups to its subgroups. These objects were introduced into consideration in a connection with an investigation of analogies between symmetries in the classical mechanics and in the relativistic one. The author introduce some new notions into consideration (for example, a weak gyrotransversal) and give a full description of left gyrogroups (and gyrogroups) in terms of transversal identities. Also he generalize a construction of a diagonal transversal and obtain a set of new examples of left gyrogroups.
format	Article
author	Kuznetsov, E.
spellingShingle	Kuznetsov, E. Gyrogroups and left gyrogroups as transversals of a special kind Algebra and Discrete Mathematics
author_facet	Kuznetsov, E.
author_sort	Kuznetsov, E.
title	Gyrogroups and left gyrogroups as transversals of a special kind
title_short	Gyrogroups and left gyrogroups as transversals of a special kind
title_full	Gyrogroups and left gyrogroups as transversals of a special kind
title_fullStr	Gyrogroups and left gyrogroups as transversals of a special kind
title_full_unstemmed	Gyrogroups and left gyrogroups as transversals of a special kind
title_sort	gyrogroups and left gyrogroups as transversals of a special kind
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2003
url	http://dspace.nbuv.gov.ua/handle/123456789/155728
citation_txt	Gyrogroups and left gyrogroups as transversals of a special kind / E. Kuznetsov // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 3. — С. 53–81. — Бібліогр.: 15 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT kuznetsove gyrogroupsandleftgyrogroupsastransversalsofaspecialkind
first_indexed	2025-07-14T07:57:51Z
last_indexed	2025-07-14T07:57:51Z
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fulltext	Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 3. (2003). pp. 54 – 81 c© Journal “Algebra and Discrete Mathematics” Gyrogroups and left gyrogroups as transversals of a special kind Eugene Kuznetsov Communicated by A. I. Kashu Abstract. In this article we study gyrogroups and left gy- rogroups as transversals in some suitable groups to its subgroups. These objects were introduced into consideration in a connection with an investigation of analogies between symmetries in the clas- sical mechanics and in the relativistic one. The author introduce some new notions into consideration (for example, a weak gyro- transversal) and give a full description of left gyrogroups (and gy- rogroups) in terms of transversal identities. Also he generalize a construction of a diagonal transversal and obtain a set of new ex- amples of left gyrogroups. 1. Introduction At the first time the concepts of a gyrogroup and a gyrocommutative gy- rogroup were introduced into consideration in [14] in a connection with an investigation of analogies between symmetries in the classical mechan- ics and in the relativistic one. In [14, 15] the elementary properties of gyrogroups were established and it was shown that they are left special loops. In [4] the concept of the gyrogroup was generalized and it was intro- dused a notion of a left gyrogroup; also in [4, 3, 5, 6] these objects were considered as transversals (gyrotransversals) in some groups to their sub- groups. In the present work the research of above-mentioned concepts is pro- ceeded. In §1 the necessary definitions are introduced, among which the 2000 Mathematics Subject Classification: 20N05, 20N15. Key words and phrases: loop, group, transversal, automorphism, gyrogroup. Jo u rn al A lg eb ra D is cr et e M at h .E. Kuznetsov 55 concepts of a weak gyrotransversal and a middle Bol loop deserves the special attention. The elementary properties of these objects are proved. In §2 it is shown that the left gyrogroups is exactly weak gyrotransver- sals in some groups; and some its properties are established. In §3 and §4 the transversals in groups are investigated, for which the transversal operations are gyrogroups and commutative gyrogroups, respectively. In §5, proceeding from a definition of semidirect product of a left loop and a suitable group (see [12, 11]), it is shown that weak gyrotransversals (gyro- transversals) are obtained by the natural way in the semidirect products of the left gyrogroups (gyrogroups) and the suitable groups. At last, in §6 the generalization of a construction of the diagonal transversals (see [4]) is given. 2. Necessary definitions, notations and preliminary state- ments Definition 1. [2] A system 〈E, ·〉 is called a left (right) quasigroup if the equation a · x = b (y · a = b) has an unique solution in the set E for anyone given a, b ∈ E. The system 〈E, ·〉 is called a quasigroup if it is both left and right quasigroup simultaneously. A left (right) quasigroup 〈E, ·〉 is called a left (right) loop if there exists the element e ∈ E such that e · x = x ( x · e = x, respectively). This element e ∈ E is called a left (right) unit. The system 〈E, ·〉 is called a loop if it is both left and right loop simultaneously (in this case e · x = x · e = x ∀x ∈ E, and this element e ∈ E is called a unit of the loop 〈E, ·〉). Definition 2. [1, 10] Let 〈G, ·, e〉 be a group and 〈H, ·, e〉 be its own subgroup. A complete set T = {ti}i∈E of representatives of the left (right) cosets of the group G to its subgroup H (exactly one representative from each coset) is called a non-reduced left (right) transversal in G to H. If the non-reduced left (right) transversal T satisfy the condition e = t1 ∈ T , then the set T = {ti}i∈E is called a left (right) transversal in G to H. The left transversal T in G to H is called a two-sided transversal in G to H, if it is a right transversal in G to H simultaneously. For every left transversal T = {ti}i∈E in G to H it is possible to define correctly a following operation on the set E (a transversal operation): x (T ) · y = z def ⇐⇒ txty = tzh, h ∈ H. (1) Lemma 1. The following statements are true: Jo u rn al A lg eb ra D is cr et e M at h .56 Gyrogroups and left gyrogroups as transversals... 1. If T is a non-reduced left (right) transversal in G to H, then the system 〈E, (T ) · 〉 is a left (right) quasigroup with the right (left) unit 1; 2. If T is a left (right) transversal in G to H, then the system 〈E, (T ) · 〉 is a left (right) loop with the unit 1. Proof. The proof is similar to the proof of Lemma 1 from [10]. Definition 3. A left transversal T in G to H is called a loop transversal if the transversal operation 〈E, (T ) · , 1〉 is a loop. Lemma 2. For every left transversal in G to H the following statements are equivalent: 1. T is a loop transversal in G to H; 2. ∀π ∈ G the set T π = πTπ−1 is a left transversal in G to H; 3. ∀π ∈ G the set T is a left transversal in G to πHπ−1. Proof. The proof it can see in [1]. For every left transversal T in G to H we shall denote: lx,y def = t−1 x·ytxty ∈ H. Definition 4. [13] A left multiplication group of a left quasigroup 〈E, ·〉: LM ( 〈E, ·〉) def = 〈La\|La (x) = a · x, a ∈ E〉. A left inner mappings group of a left loop 〈E, ·, 1〉: LI ( 〈E, ·, 1〉) def = {α ∈ LM ( 〈E, ·, 1〉) \|α (1) = 1}. It is known (see [13]) that LI ( 〈E, ·, 1〉) = 〈lx,y\|x, y ∈ E〉. Definition 5. A left transversal T = {ti}i∈E in G to H is called a 1. weak gyrotransversal, if the following conditions hold: (a) T is a two-sided transversal in G to H; Jo u rn al A lg eb ra D is cr et e M at h .E. Kuznetsov 57 (b) LI(〈E, (T ) · , 1〉) ⊆ NG (T ), i.e. ∀h ∈ LI(〈E, (T ) · , 1〉) it is true that hTh−1 ⊆ T ; 2. gyrotransversal [4, 3], if the following conditions hold: (a) ∀ti ∈ T it is true that t−1 i ∈ T ; (b) H ⊆ NG (T ), i.e. ∀h ∈ H it is true that hTh−1 ⊆ T . Definition 6. [4]A system 〈E, ·〉 is called a left gyrogroup, if the fol- lowing conditions hold: 1. In the set E there exists an element 1 such that 1 · x = x ∀x ∈ E. 2. ∀x ∈ E there exists an element −1x ∈ E such that −1x · x = 1. 3. ∀a, b, z ∈ E the following identity holds: a · (b · z) = (a · b) · αa,b (z) , where αa.b ∈ Aut ( 〈E, ·〉) is called a gyroautomorphism. Remark 1. A left gyrogroup 〈E, ·, 1〉 is a left loop, i.e. the equation a · x = b has the unique solution in E for every fixed a, b ∈ E. Really, let a · x = b. Then for left opposite −1a to a ∈ E we have: −1a · b = −1a · (a · x) = ( −1a · a ) · α−1a,a (x) = = 1 · α−1a,a (x) = α−1a,a (x) , i.e. x = α−1 −1a,a ( −1a · b ) . Definition 7. [14, 4, 3, 15] A left gyrogroup 〈E, ·, 1〉 is called a gy- rogroup, if ∀a, b ∈ E the following condition holds: αa,b ≡ αa·b,b. Definition 8. [14, 4, 3, 15] A gyrogroup 〈E, ·, 1〉 is called a gyrocom- mutative gyrogroup, if ∀a, b ∈ E the following condition holds: a · b = αa,b (b · a) . Jo u rn al A lg eb ra D is cr et e M at h .58 Gyrogroups and left gyrogroups as transversals... Below we shall consider a group G as its permutation representation Ĝ by the left cosets to its subgroup H. If T = {tx}x∈E is a left transversal in G to H, we define [10]: ĝ (x) = y def ⇐⇒ gtxH = tyH. (2) It is known [8] that if CoreG(H) = ∩ g∈G gHg−1 = {e}, then Ĝ ∼= G; below we shall propose that CoreG(H) = {e}. Lemma 3. Let T = {tx}x∈E is a non-reduced left transversal in G to H and 〈E, (T ) · , 1〉 is the transversal operation. Then the following formulas are true: 1. ∀h ∈ H: ĥ (1) = 1; 2. ∀x, y ∈ E: t̂x (y) = x (T ) · y, t̂x (1) = x, t̂−1 x (1) = x (T ) \ 1, t̂−1 x (y) = x (T ) \ y, t̂−1 x (x) = 1. 3. If T = {tx}x∈E is a left transversal in G to H, then also the fol- lowing identity is fulfilled: t̂1 (x) = x. Proof. The proof is similar to the proof of Lemma 4 from [10]. Lemma 4. Let T = {tx}x∈E be a non-reduced left transversal in G to H and 〈E, (T ) · , 1〉 be its transversal operation. Then the following statements are equivalent: 1. T is a non-reduced two-sided transversal in G to H; 2. The equation x (T ) · a = 1 has an unique solution in E for every a ∈ E; 3. A set T−1 = { t−1 x } x∈E is a non-reduced two-sided transversal in G to H. Jo u rn al A lg eb ra D is cr et e M at h .E. Kuznetsov 59 Proof. 1) ←→ 2). The proof is similar to the proof of Lemma 7 from [10]. 1) ←→ 3). Since ∀x ∈ E it is true that (Htx)−1 = t−1 x H, (txH)−1 = Ht−1 x , so if T is a non-reduced left (right) transversal in G to H then T−1 is a non-reduced right (left) transversal in G to H, and vice versa. Definition 9. [2] A left (right) quasigroup 〈E, ·〉 is called a special quasigroup at the left (at the right), if ∀x, y ∈ E lx,y = L−1 x·yLxLy ∈ Aut ( 〈E, ·〉) ( rx,y = R−1 x·yRyRx ∈ Aut( 〈E, ·〉), respectively). Definition 10. [2] A left loop 〈E, ·〉 is called a left Bol loop, if the following identity (left Bol identity) is fulfilled ∀x, y, z ∈ E: x (y (xz)) = (x (yx)) z. Lemma 5. A left Bol loop 〈E, ·, 1〉 satisfies the following properties: 1. the left inverse property, i.e. ∀x, y ∈ E: −1x · (x · z) = z, where −1x · x = 1; 2. −1x = x−1, i.e. the left and the right inverse elements to an element x ∈ E coincide; 3. the left alternation, i.e. ∀x, y ∈ E: x · (x · y) = (x · x) · y; 4. the solution of the equation a ·x = b is x = a−1 · b, and the solution the equation y · a = b is y = a−1 · ( (a · b) · a−1 ) , i.e. left Bol loop 〈E, ·, 1〉 is a loop. Proof. The proof it can see in [2], chapter 6. Definition 11. An operation 〈E, ·〉 is called a middle Bol loop [2], if the following identity holds: x · ((yz) \x) = (x/z) · (y\x) , where ”\” and ”/” are left and right divisions in 〈E, ·〉, respectively. Lemma 6. Let 〈E, ·〉 is a middle Bol loop. Then the following statements are true: 1. 〈E, ·〉 is a loop with some unit 1; Jo u rn al A lg eb ra D is cr et e M at h .60 Gyrogroups and left gyrogroups as transversals... 2. The left inverse element −1x and the right inverse element x−1 to an element x ∈ E coincide: −1x = x−1; 3. If 〈E, ·, 1〉 is a left Bol loop and ”/” is the right inverse operation to the operation ” · ”, then the operation x ◦ y = x/y−1 is a middle Bol loop 〈E, ◦, 1〉, and everyone middle Bol loop can be obtained in a similar way from some left Bol loop. Proof. The proof it can see in [7]. 3. Left gyrogroups as weak gyrotransversals Lemma 7. Let T be a left transversal in G to H. Then the following statements are equivalent: 1. T is a weak gyrotransversal in G to H; 2. The transversal operation 〈E, (T ) · , 1〉 is a left gyrogroup. Proof. 1) −→ 2). Let T be a left transversal in G to H and T be a weak gyrotransversal in G to H, i.e. the two following conditions hold: 1. T is a two-sided transversal in G to H; 2. ∀h ∈ LI (〈E, (T ) · , 1〉) it is true that hTh−1 ⊆ T . Let us show that the conditions 1) - 3) from Definition 6 are fulfilled for the operation 〈E, (T ) · , 1〉. The condition 1) is fulfilled automatically for anyone left transversal in G to H (see Lemma 1). The condition 2) follows from the Condition 1 and Lemma 4. Valid Condition 2 ∀x ∈ E and ∀h ∈ LI (〈E, (T ) · , 1〉) it is true that htxh−1 = tψ(x). (3) In virtue Lemma 3 we have: ψ (x) = t̂ψ(x) (1) = ĥt̂xĥ−1 (1) = ĥt̂x (1) = ĥ (x) , i.e. the equality (3) may be rewritten as: htxh−1 = t ĥ(x). (4) Jo u rn al A lg eb ra D is cr et e M at h .E. Kuznetsov 61 Let us show that ∀h ∈ LI (〈E, (T ) · , 1〉) a mapping αh : x → ĥ (x) is an automorphism of the operation 〈E, (T ) · , 1〉. We have ∀x, y ∈ E: txty = tx·ylx,y, where lx,y = t−1 x·ytxty ∈ LI(〈E, (T ) · , 1〉). Then in virtue of (4) ∀h ∈ LI(〈E, (T ) · , 1〉) it is true that htxh−1htyh −1 = htx·yh −1hlx,yh −1, t ĥ(x)tĥ(y) = t ĥ(x·y)h ′, where h′ ∈ LI (〈E, (T ) · , 1〉), and ĥ (x) (T ) · ĥ (y) = ĥ(x (T ) · y), i.e. αh is an automorphism of the operation 〈E, (T ) · , 1〉. At last ∀a, b, z ∈ E we have in virtue of (4): tatbtz = ta·bla,btz = ta·bla,btzl −1 a,b la,b = = ta·btl̂a,b(z)la,b = t(a·b)·l̂a,b(z)la·b,l̂a,b(z)la,b, and, on the other hand tatbtz = tatb·zlb,z = ta·(b·z)la,b·zlb,z. So we obtain ta·(b·z)la,b·zlb,z = t(a·b)·l̂a,b(z)la·b,l̂a,b(z)la,b. According to the definition of a left transversal it means that a (T ) · (b (T ) · z) = (a (T ) · b) (T ) · l̂a,b (z) , where αla,b = l̂a,b is an automorphism of the operation 〈E, (T ) · , 1〉 (as it was shown above). The condition 3) of Definition 6 is fulfilled. 2) −→ 1). Let T = {tx}x∈E be a left transversal in G to H and the operation 〈E, (T ) · , 1〉 be a left gyrogroup. Then the conditions 1)-3) from Jo u rn al A lg eb ra D is cr et e M at h .62 Gyrogroups and left gyrogroups as transversals... Definition 6 are fulfilled for the operation 〈E, (T ) · , 1〉. Let us show that T is a weak gyrotransversal. By virtue of condition 2) and Lemma 4 the transversal T is a two-sided transversal, i.e. the item 1a) of Definition 5 holds. By virtue of condition 3) it is true that ∀a, b, z ∈ E: t̂at̂b (z) = t̂a·b (αa,b (z)) , i.e. la,b = αa,b is an automorphism of the operation 〈E, (T ) · , 1〉. Then LI(〈E, (T ) · , 1〉) ⊆ Aut(〈E, (T ) · , 1〉) (5) Now let h ∈ LI(〈E, (T ) · , 1〉) and we shall consider the expression ( htkh −1 ) ∀x ∈ E. Since ∀x ∈ E htxh−1 ∈ G then htxh−1 = tuh1 (6) for some u ∈ E and h1 ∈ H. Valid Lemma 3 we have: u = t̂u (1) = t̂uĥ1 (1) = ĥt̂xĥ−1 (1) = ĥt̂x (1) = ĥ (x) . So (6) may be rewritten as htxh−1 = t ĥ(x)h 1. (7) Further we have ∀x, y ∈ E: txty = tx·ylx,y, htxh−1htyh −1 = htx·yh −1hlx,yh −1. In virtue of (7) we obtain: t ĥ(x)h1tĥ(y)h2 = t ĥ(x·y)h3 · hlx,yh −1, h1, h2, h3 ∈ H. Again in virtue Lemma 3 we have t̂ ĥ(x)ĥ1t̂ĥ(y)ĥ2 (1) = t̂ ĥ(x·y)ĥ3ĥ l̂x·yĥ −1 (1) , ĥ (x) (T ) · ĥ1 ( ĥ (y) ) = ĥ(x (T ) · y). (8) Since h ∈ LI(〈E, (T ) · , 1〉) then in virtue of (5) we have ĥ(x (T ) · y) = ĥ (x) (T ) · ĥ (y) . Jo u rn al A lg eb ra D is cr et e M at h .E. Kuznetsov 63 Substituting the last equality in (8), we obtain ĥ (x) (T ) · ĥ1 ( ĥ (y) ) = ĥ (x) (T ) · ĥ (y) ∀x, y ∈ E. Since T is a left transversal in G to H then system 〈E, (T ) · , 1〉 is a left quasigroup; therefore we receive ĥ1 ( ĥ (y) ) = ĥ (y) ∀y ∈ E. Since the mapping ĥ (y) is a permutation on the set E then we have: ĥ1 (z) = z ∀z ∈ E, i.e. ĥ1 = id and so h1 = e. Then according to (7), we receive that ∀h ∈ LI(〈E, (T ) · , 1〉) and ∀x ∈ E it is true that htxh−1 ∈ T, i.e. the item 1b) of Definition 5 is fulfilled. Then the transversal T is a weak gyrotransversal. Corollary 11. If a left transversal T in G to H is a gyrotransversal then the transversal operation 〈E, (T ) · , 1〉 is a left gyrogroup. Lemma 8. Let T be a weak gyrotransversal in G to H (i.e. the transver- sal operation 〈E, (T ) · , 1〉 is a left gyrogroup). Then the transversal opera- tion 〈E, (T ) · , 1〉 satisfies the following properties: 1. a (T ) · b = a (T ) · c ⇔ b = c (left cansellation); 2. The element 1 ∈ E is the unique unit for the operation 〈E, (T ) · , 1〉; 3. ∀a ∈ E there exist an unique left inverse element −1a (−1a (T ) · a = 1) and an unique right inverse element −1a (a (T ) · a−1 = 1). 4. If αa,b is a gyroautomorphism then αa,b (z) = (a (T ) · b)\(a (T ) · (b (T ) · z)), α−1 a,b (z) = b\(a\((a (T ) · b) (T ) · z)), and, as a corollary, α0,a = αa,0 = id. Jo u rn al A lg eb ra D is cr et e M at h .64 Gyrogroups and left gyrogroups as transversals... 5. It is true that ∀x, y ∈ E: x (T ) · (x−1(T ) · y) = ϕx (y) , −1x (T ) · (x (T ) · y) = ψx (y) , where ϕx and ψx are some automorphisms of operation 〈E, (T ) · , 1〉. Proof. 1) and 2) follow from Lemma 1. 3) follows from Lemma 1 and Lemma 4 (item b)). 4). According to the definition of a left gyrogroup a · (b · z) = (a · b) · αa,b (z) . On the other hand, since T is a left transversal then tatb = ta·bla,b, and so ∀z ∈ E a · (b · z) = t̂a (b · z) = t̂at̂b (z) = t̂a·b l̂a,b (z) = (a · b) · l̂a,b (z) . So αa,b = ĥa,b. Then, according Lemma 3 αa,b (z) = l̂a,b (z) = t̂−1 a·b t̂at̂b (z) = (a · b) \ (a · (b · z)) , α−1 a,b (z) = l̂−1 a,b (z) = t̂−1 b t̂−1 a t̂a·b (z) = b\ (a\ (a · b) · z) . 5). From the item 4) it follows that x · ( x−1 · y ) = ( x · x−1 ) · αx,x−1 (y) = ϕx (y) , −1x · (x · y) = ( −1x · x ) · α−1x,x (y) = ψx (y) . Remark 2. In a left gyrogroup the left inverse property may not be fulfilled. The example it can see in [9], page 317-318. 4. Gyrogroups as loop transversals of a special kind Lemma 9. Let T = {tx}x∈E be a weak gyrotransversal in G to H. Then the following statements are equivalent: 1. The transversal operation 〈E, (T ) · , 1〉 is a gyrogroup; 2. ∀x ∈ E: txTtx ⊆ T ; 3. The transversal operation 〈E, (T ) · , 1〉 is a left Bol loop. Jo u rn al A lg eb ra D is cr et e M at h .E. Kuznetsov 65 Proof. 1) −→ 2). Let T = {tx}x∈E be a weak gyrotransversal in G to H and transversal operation 〈E, (T ) · , 1〉 is a gyrogroup. Then ∀x, y ∈ E: αx,y = αx·y,y. In virtue of Definition 6 and Lemma 3 we have: t−1 x·ytxty = t−1 (x·y)·ytx·yty, t−1 x·ytxt−1 x·y = t−1 (x·y)·y, tx·yt −1 x tx·y = t(x·y)·y (9) If y = x−1 then from (9) we obtain that t−1 x = e · t−1 x · e = tx−1 , (10) i.e. (9) may be rewritten as tx·ytx−1tx·y = t(x·y)·y. (11) Since 〈E, (T ) · , 1〉 is a left loop, then the following replacement is correct: { x−1 = u x · y = v −→ y = x\v = (1/u) \v = ( u−1 ) \v. Then (11) may be rewritten as: ∀u, v ∈ E tvtutv = tv·((−1u)\v), i.e. tvTtv ⊆ T ∀v ∈ E. 2) −→ 3). Let T be a left transversal in G to H and ∀x ∈ E txTtx ⊆ T, Then ∀x, y ∈ E there exists a permutation αx (y) such that txtytx = tαx(y). (12) In virtue Lemma 3 we have αx (y) = t̂αx(y) (1) = t̂xt̂y t̂x (1) = t̂xt̂y (x) = x · (y · x) . Then (12) may be rewritten as txtytx = tx·(y·x). Jo u rn al A lg eb ra D is cr et e M at h .66 Gyrogroups and left gyrogroups as transversals... Again applying Lemma 3, we obtain: x · (y · (x · z)) = x · ( y · t̂x (z) ) = x · t̂y t̂x (z) = = t̂xt̂y t̂x (z) = t̂x·(y·x) (z) = (x · (y · x)) · z, i.e. the left Bol identity is fulfilled for the operation 〈E, (T ) · , 1〉. Then 〈E, (T ) · , 1〉 is a left Bol loop. 3) −→ 1). Let T be a weak gyrotransversal and 〈E, (T ) · , 1〉 be a left Bol loop. Then 〈E, (T ) · , 1〉 is a left gyrogroup and the left Bol identity holds: x · (y · (x · z)) = (x · (y · x)) · z. Then in virtue Lemma 3 we obtain ∀z ∈ E: t̂xt̂y t̂x (z) = x · t̂y t̂x (z) = x · ( y · t̂x (z) ) = = x · (y · (x · z)) = (x · (y · x)) · z = t̂x·(y·x) (z) i.e. txtytx = tx·(y·x). (13) Besides for a left Bol loop 〈E, (T ) · , 1〉 it is true, that for every element x ∈ E the left inverse element −1x coincides with the right inverse element x−1: −1x = x−1. Also we know that a left Bol loop 〈E, (T ) · , 1〉 is a left IP -loop, i.e. −1x · (x · y) = x · (( −1x ) · y ) = y. (14) Let us do a replacement: { y−1 = u y · x = v −→ x = y−1 · (y · x) = u · v. Then (13) may be rewritten as: ∀u, v ∈ E tu·vtu−1tu·v = t(u·v)·v. (15) Valid (14) we obtain ∀x, y ∈ E: y = x · ( x−1 · y ) , x\y = x−1 · y, t̂−1 x (y) = t̂x−1 (y) , t−1 x = tx−1 . Jo u rn al A lg eb ra D is cr et e M at h .E. Kuznetsov 67 By virtue of the last equality we obtain from (15): ∀u, v ∈ E tu·vt −1 u tu·v = t(u·v)·v, t−1 u·vtut−1 u·v = t−1 (u·v)·v, t−1 u·vtu = t−1 (u·v)·vtu·v, t−1 u·vtutv = t−1 (u·v)·vtu·vtv, αu,v = αu·v,v, i.e. operation 〈E, (T ) · , 1〉 is a gyrogroup. Lemma 10. Let T = {tx}x∈E be a left transversal in G to H and 〈E, (T ) · , 1〉 be the transversal operation. Then the following statements are equivalent: 1. The system 〈E, (T ) · , 1〉 is a gyrogroup; 2. T is a two-sided transversal in G to H and two following conditions hold: (a) ∀x ∈ E: txTtx ⊆ T ; (b) ∀h ∈ LI(〈E, (T ) · , 1〉): hTh−1 ⊆ T . Proof. The proof is an evident corollary of Lemmas 7 and 9. Remark 3. Since a left Bol loop is a loop then the transversal T from Lemmas 9 and 10 is a loop transversal. Lemma 11. [15] In every gyrogroup 〈E, ·, 1〉 the following properties are fulfilled: 1. ∀a ∈ E there exists the unique element a−1 ∈ E such that a·a−1 = a−1 · a = 1; 2. αa,a−1 = αa−1,a = αa,a = id; 3. ∀a, b ∈ E: αa,b (z) = (a · b)−1 · (a · (b · z)) , α−1 a,b (z) = b−1 · ( a−1 · ((a · b) · z) ) . 4. αa,b ( b−1 · a−1 ) = (a · b)−1; Jo u rn al A lg eb ra D is cr et e M at h .68 Gyrogroups and left gyrogroups as transversals... 5. α−1 a,b = αa−1,a·b = αb,a·b = αb,a; 6. αa,b = αa·b,a−1 = αa,b·a; 7. (a · b) · c = a · (b · αb,a (c)); 8. The solution of the equation x · a = b is: x = b · (αb,a (a))−1 = b · α−1 a,b ( a−1 ) . Proof. 1). Since a gyrogroup is a left Bol loop then in virtue Lemma 5, item 2) it is true that −1x = x−1, so x · x−1 = x−1 · x = 1. 2). In virtue Lemma 5, items 1) and 2) for a left Bol loop it is true that x−1 · (x · z) = x · ( x−1 · z ) = z ∀x, z ∈ E. Therefore αa,a−1 (z) = ( a · a−1 ) \ ( a · ( a−1 · z )) = a · ( a−1 · z ) = z, i.e. αa,a−1 = id. Similarly, αa−1,a = id. Further we have in virtue Lemma 5, item 3) αa,a (z) = (a · a) \ (a · (a · z)) == (a · a) \ ((a · a) · z) = z, i.e. αa,a = id. 3). By the definition 6 a · (b · z) = (a · b) · αa,b (z) , so in virtue Lemma 5, item 3) we have αa,b (z) = (a · b)−1 · (a · (b · z)) . (16) Making the replacement z = α−1 a,b (u), we obtain from (12): u = (a · b)−1 · (a · (b · α−1 a,b (u) )), (17) and again using Lemma 5, item 3) we obtain: α−1 a,b (u) = b−1 · (a−1·((a · b) · u)). 4). Using item 3) of present Lemma and Lemma 5, item 3) we obtain: αa,b ( b−1 · a−1 ) = (a · b)−1 · (a·(b·(b−1 · a−1))) = = (a · b)−1 · ( a · a−1 ) = (a · b)−1 . Jo u rn al A lg eb ra D is cr et e M at h .E. Kuznetsov 69 5). According to item 3) of present Lemma α−1 a,b (z) = b−1 · (a−1 · ((a · b) · z)). (18) Let us do a replacement { a−1 = c a · b = d −→ { c = a−1 c · d = a−1 · (a · b) = b Then (18) may be rewritten as α−1 a,b (z) = (c · d)−1 · (c·(d · z)) = αc,d (z) = αa−1,a·b (z) , i.e. α−1 a,b ≡ αa−1,a·b. Further in virtue Lemma 9 if a transversal T = {tx}x∈E corresponds to a gyrogroup 〈E, ·, 1〉 = 〈E, (T ) · , 1〉, then it is true that txtytx = tx·(y·x), ∀x, y ∈ E. Then we have tx·(y·x) = txtytx = txty·xαy,x = tx·(y·x)αx,y·xαy,x. So we obtain αx,y·xαy,x = id, i.e. α−1 a,b = αb,a·b. (19) At last since by the Definition 6 for every gyrogroup it is true that αa,b = αa·b,b, then from (19) we obtain: α−1 a·b,b = α−1 a,b = αb,a·b. (20) Making the replacement { a · b = c b = d we obtain from (20) ∀c, d ∈ E: α−1 c,d = αd,c. Jo u rn al A lg eb ra D is cr et e M at h .70 Gyrogroups and left gyrogroups as transversals... 6). We have from item 5) of present Lemma αa,b = α−1 a−1,a·b = αa·b,a−1 . (21) By virtue of the last equality in the item 5) we have αb, a·b = αb,a, i.e. αa,b = αa,b·a. 7). By virtue of items 3) and 5) of present Lemma and Lemma 5, item 1) we have: (a · b) · c = a · (b · α−1 a,b (c) ) = a · (b · αb,a (c)) . 8). According to items 1) and 3) of present Lemma and Lemma 5, item 4) the solution of the equation x · a = b is x = a−1 · ( (a · b) · a−1 ) = b · ( b−1 · ( a−1 · ( (a · b) · a−1 ))) = (22) = b · α−1 a,b ( a−1 ) = b · αb,a ( a−1 ) . But since 1 = αb,a ( a−1 · a ) = αb,a ( a−1 ) · αb,a (a) , (23) then we obtain from (22): x = b · αb,a ( a−1 ) = b · (αb,a (a))−1 . It is very interesting to investigate operations, which are inverse ones to a gyrogroup operation 〈E, ·, 1〉. The left inverse operation coincides with operation 〈E, ·, 1〉 (because of the left Bol loop 〈E, ·, 1〉 is a LIP - loop). Let us study the right inverse operation. We can define the following operations on a set E (see [15]): a ⊕ b def = a · αa,b−1 (b) , (24) a ¯ b def = a ⊕ b−1. Lemma 12. Let 〈E, ·, 1〉 be a gyrogroup. Then the following statements are true: 1. a ¯ b = a/b, a ⊕ b = a/b−1, where “/” is a right division in the gyrogroup 〈E, ·, 1〉, a · b = a // b−1, where “//” is a right division in a system 〈E,⊕, 1〉; Jo u rn al A lg eb ra D is cr et e M at h .E. Kuznetsov 71 2. a ⊕ αa,b (b) = a · b; 3. The system 〈E,⊕, 1〉 is a loop with the unit 1, and ∀x ∈ E the left and right inverse elements to an element x in 〈E,⊕, 1〉 coincide. Moreover, both of them are equal to x−1 (where x−1 is an inverse element to an element x in 〈E, ·, 1〉); 4. (a ⊕ b)−1 = b−1 ⊕ a−1; 5. Aut (〈E,⊕, 1〉) = Aut (〈E, ·, 1〉); 6. The system 〈E,⊕, 1〉 is a middle Bol loop, i.e. the following identity holds: x ⊕ ((y ⊕ z) \ \x) = (x //z) ⊕ (y \\x) , where “\\” and “//” are left and right division in 〈E,⊕, 1〉, respec- tively. Proof. 1). According to (23), (24) and Lemma 11, item 8), we obtain a ¯ b = a ⊕ b−1 = a · αa,b ( b−1 ) = a · (αa,b (b))−1 = a/b. Then a ⊕ b = a ¯ b−1 = a/b−1. Further we have (a · b) ⊕ b−1 = (a · b)/b = a, i.e. a · b = a//b−1, where “//” is a right division in 〈E,⊕, 1〉. 2). From item 1) and (23) it follows that (a · b) // (αa,b (b)) = (a · b) · (αa,b (b))−1 = = (a · b) · αa,b ( b−1 ) = a · ( b · b−1 ) = a, i.e. a ⊕ αa,b (b) = a · b. 3). In virtue of the item 1) the system 〈E,⊕, 1〉 is an inverse operation to the loop 〈E, ·, 1〉, therefore it is a quasigroup. Further we have ∀x ∈ E: 1 ⊕ x = 1/x−1 = x, x ⊕ 1 = x/1−1 = x, Jo u rn al A lg eb ra D is cr et e M at h .72 Gyrogroups and left gyrogroups as transversals... i.e. 〈E,⊕, 1〉 is a loop. At last, x ⊕ x−1 = x/ ( x−1 )−1 = x/x = 1, x−1 ⊕ x = x−1/x−1 = 1. 4). We have ∀a, b ∈ E: (a ⊕ b)−1 = ( a/b−1 )−1 , b−1 ⊕ a−1 = b−1/a. But a = c · b−1 for some c ∈ E, therefore (a ⊕ b)−1 = (( c · b−1 ) /b−1 )−1 = c−1 = ( c−1 · a ) /a = b−1/a = b−1⊕a−1. 5). According to the item 1), we have a ⊕ b = a/b−1, a · b = a//b−1. Then every automorphism α of the operation 〈E, ·, 1〉 will be an automor- phism of the inverse operation 〈E, /〉, and so α will be an automorphism of the operation 〈E,⊕, 1〉; and vice versa. 6). It is an evident corollary of Lemma 6, item 3). Let us note also the folowing identities (x//y)−1 = z−1\\x−1, (x\\y)−1 = y−1//x−1. 5. Gyrocommutative gyrogroups Lemma 13. Let T = {tx}x∈E be a left transversal in G to H such that the transversal operation 〈E, (T ) · , 1〉 is a gyrogroup. Then the following statements are equivalent: 1. 〈E, (T ) · , 1〉 is a gyrocommutative gyrogroup; 2. ∀x, y ∈ E : (x · y) · (x · y) = x · (y · (y · x)) , - the Bruck identity; 3. ∀x, y ∈ E : txt2ytx = t2x·y; 4. ∀x, y ∈ E : (x · y)−1 = x−1 · y−1, - automorphic inverse property. Jo u rn al A lg eb ra D is cr et e M at h .E. Kuznetsov 73 Proof. Let the conditions of Lemma hold; then 〈E, (T ) · , 1〉 is a left Bol loop. 1) −→ 2). Let 〈E, (T ) · , 1〉 is a gyrocommutative gyrogroup. Then by the definition 8 ∀x, y ∈ E: x · y = αx,y (y · x) . Then by the definition 6 of the automorphism αx,y and in virtue Lemma 5 we have: x · y = (x · y)−1 · (x · (y · (y · x))) , (x · y) · (x · y) = x · (y · (y · x)) . 2) −→ 3). Let the following identity holds ∀x, y ∈ E: (x · y) · (x · y) = x · (y · (y · x)) . Then in virtue Lemmas 5 and 9 we have ∀x, y ∈ E: txt2ytx = txty·ytx = tx·((y·y)·x) = tx·(y·(y·x)) = t(x·y)·(x·y)=t2x·y. 3) −→ 4). Let ∀x, y ∈ E txt2ytx = t2x·y. Then in virtue Lemma 5 we have: txtytytx = tx·ytx·y, lx,y = t−1 x·ytxty = tx·yt −1 x t−1 y , l̂x,y(1) = t̂x·y t̂ −1 x t̂−1 y (1), 1 = (x · y) · ( x−1 · y−1 ) , (x · y)−1 = x−1 · y−1. 4) −→ 1). Let ∀x, y ∈ E: (x · y)−1 = x−1 · y−1. Then in virtue Lemma 11, item 4) we have (x · y)−1 = αx·y ( y−1 · x−1 ) = αx,y((y · x)−1). Then we obtain 1 = αx,y (1) = αx,y((y · x) · (y · x)−1) = = αx,y (y · x) · αx,y((y · x)−1) = αx,y (y · x) · (x · y)−1 , Jo u rn al A lg eb ra D is cr et e M at h .74 Gyrogroups and left gyrogroups as transversals... i.e. αx,y (y · x) = x · y, and the system 〈E, (T ) · , 1〉 is a gyrocommutative gyrogroup. Lemma 14. Let T = {tx}x∈E is a left transversal in G to H and 〈E, (T ) · , 1〉 is a transversal operation. Then the following statements are equivalent: 1. 〈E, (T ) · , 1〉 is a gyrocommutative gyrogroup; 2. T is a two-sided transversal in G to H and the following three con- ditions hold: (a) ∀x ∈ E : txTtx ⊆ T ; (b) ∀h ∈ LI (〈E, (T ) · , 1〉): hTh−1 ⊆ T ; (c) ∀x ∈ E : txt2ytx = t2x·y. Proof. The proof is an evident corollary from Lemmas 10 and 13. Lemma 15. If 〈E, (T ) · , 1〉 is a gyrocommutative gyrogroup then the opera- tion “⊕” (determined in (24)) satisfies the following properties: ∀x, y ∈ E 1. x ⊕ y = y ⊕ x; 2. x//y = y\\x. Proof. 1). According Lemma 12 we have x ⊕ y = x/y−1, so it is necessary to prove that ∀x, y ∈ E x/y−1 = y/x−1. But since 〈E, (T ) · , 1〉 is a loop then x = z · y−1 for some z ∈ E. In virtue Lemma 13 we obtain ( x/y−1 ) · x−1 = (( z · y−1 ) /y−1 ) · ( z · y−1 )−1 = z · ( z−1 · y ) = y, as it was required. Jo u rn al A lg eb ra D is cr et e M at h .E. Kuznetsov 75 2). According Lemma 12 we have x//y = x · y−1, then we obtain y\\x = ( x−1//y−1 )−1 = ( x−1 · y )−1 = x · y−1 = x//y. 6. Semidirect products of gyrogroups, left gyrogroups and suitable groups Let us remind a definition of a semidirect product of a left loop 〈E, ·, 1〉 and a suitable permutation group H (see [11, 12]). Definition 12. Let 〈E, ·, 1〉 be a left loop with two-sided unit 1, and H be a subgroup of the permutation group St1 (SE) such that the following conditions are fulfilled: 1. ∀a, b ∈ E: la,b = L−1 a·bLaLb ∈ H; 2. ∀a ∈ E and ∀h ∈ H: ϕ (a, h) = L−1 h(a)hLah −1 ∈ H, where La (x) = a · x is a left translation by an element a ∈ E. Then on a set E × H of pairs (u, h) it is possible to define an operation: (u, h1) ∗ (v, h2) def = ( u · h1 (v) , la,h1(v)ϕ (v, h1)h1h2 ) , (25) and an action on the set E: (u, h) (x) def = u · h (x) . (26) It is possible to show (see [11, 12]) that: 1. A system G = 〈E×H, ∗, (1, id)〉 is a group (a semidirect product of the left loop 〈E, ·, 1〉 and the group H); 2. It is true that (u, h)−1 = (h−1 (u\1) ,(LuhL h−1(u\1) )−1); 3. A set T = { (u, id) \|u ∈ E} is a left transversal in the group G to its subgroup H∗ = { (1, h) \|h ∈ H} ∼= H, and the transversal operation 〈E, (T ) · , 1〉 coincides with the operation 〈E, ·, 1〉. Jo u rn al A lg eb ra D is cr et e M at h .76 Gyrogroups and left gyrogroups as transversals... The following special case of the above-described construction will be important for us: when the left loop 〈E, ·, 1〉 is a left special loop (left Al-loop), that is LI (〈E, ·, 1〉) ⊆ H ⊆ Aut (〈E, ·, 1〉) . (27) The formula (25) of the semidirect product may be rewritten as (u, h1) ∗ (v, h2) def = ( u · h1 (v) ,la,h1(v)h1h2 ) . (28) Then all above-mentioned properties are correct and the following formula holds: (u, h)−1 = ( h−1 (u\1) ,(LuLu\1h)−1 ) . (29) Remark 4. The formula (27) coincides with the formula of a gyrosemidi- rect product of a left gyrogroup and its gyroautomorphism group (see [4, 14]). Lemma 16. Every left gyrogroup 〈E, ·, 1〉 may be represented as a weak gyrotransversal in the group 〈E × H, ∗, (1, id)〉 to a subgroup H, if H satisfies the conditions of the Definition 12. Proof. The proof obviously follows from Lemma 7 and above-mentioned properties of the semidirect product. Corollary 12. Every left gyrogroup 〈E, ·, 1〉 may be represented as a weak gyrotransversal in the group 〈E × H, ∗, (1, id)〉 to a subgroup H0, which satisfies the condition (27) (and semidirect product is defined under the formula (28)). Lemma 17. Every gyrogroup 〈E, ·, 1〉 may be represented as a gyro- transversal in the group G = 〈E × H, ∗, (1, id)〉 to its gyroautomorphism group H0 (i.e. H0 satisfies the condition (27)). Proof. (See also [4]) According the Corollary 12 a set T = { (a, id) \|a ∈ E} is a weak gyrotransversal in G to H0. Since 〈E, ·, 1〉 is a gyrogroup then in virtue Lemma 9 〈E, ·, 1〉 is a left Bol loop; therefore it satisfies the left inverse property, i.e. ∀u ∈ E LuLu\1 = id. (30) Then in virtue of (29) (u, id)−1 = (u\1, id) = ( u−1, id ) , Jo u rn al A lg eb ra D is cr et e M at h .E. Kuznetsov 77 i.e. T−1 = T. Further ∀u ∈ E and ∀h ∈ H0 in virtue of the formula (28) (1, h) ∗ (u, id) ∗ ( 1, h−1 ) = (1, h) ∗ ( u, lu,1h −1 ) = = (1, h) ∗ ( u, h−1 ) = ( h (u) , l1,h(u)hh−1 ) = (h (u) , id) , i.e. ∀u ∈ E and ∀h ∈ H0 (1, h) ∗ T ∗ (1, h)−1 = T. It means that T is a gyrotransversal in G to H0. Remark 5. A left gyrogroup may not be represented as a gyrotransversal in the group G = 〈E×H, ∗, (1, id)〉 to its gyroautomorphism group (since in a left gyrogroup, not being a gyrogroup, it is not necessarily satisfied the condition (30)). Lemma 18. A left gyrogroup 〈E, ·, 1〉 may be represented as a gyro- transversal in the group G = 〈E × H, ∗, (1, id)〉 to a group H0, which satisfies the condition (27) ⇔ 〈E, ·, 1〉 is a LIP -loop. Proof. The proof is evident, because a left gyrogroup 〈E, ·, 1〉 is always a weak gyrotransversal in the group G = 〈E×H, ∗, (1, id)〉 to the subgroup H0 (see a Corollary 12), and the condition (30) is equivalent to a definition of LIP -loop. 7. Generalized diagonal transversals Definition 13. Let K be a group, G be a semidirect product G = K h Inn (K) , where Inn (K) = { αn\|αn (x) = kxk−1, k, x ∈ K } is a group of internal automorphisms of the group K. Then a generalized diagonal transversal Dm of degree m is a set Dm = {(k, αm k ) \|k ∈ K}. (31) We shall denote Dm (K) = (k, αm k ) . (32) A diagonal transversals, which were investigated in [4, 5, 6, 3], are obtained in a case when m = 1. Jo u rn al A lg eb ra D is cr et e M at h .78 Gyrogroups and left gyrogroups as transversals... Lemma 19. The generalized diagonal transversal Dm of degree m is a gyrotransversal in G to H = Inn(K). Proof. For every element (k, αh) ∈ G (where k, h ∈ K) we have: (k, αh) = (k, αm k ) · (1, αk−mh) , and this decomposition is an unique one. It means that the set Dm is a left transversal in G to H. Further we have: (Dm (k))−1 = (k, αm k )−1 = ( α−m k ( k−1 ) , α−m k ) = Dm ( k−1 ) , i.e. (Dm)−1 ≡ Dm. Also we obtain: (1, αh)Dm (k) (1, αh)−1 = (1, αh) (k, αm k ) (1, αh−1) = = (1, αh) (k, αkmh−1) = (αh (k) ,αhkmh−1) = = ( αh (k) ,ααh(km) ) = ( αh (k) ,α(αh(k))m ) = = (αh (k) ,αm αh(k)) = Dm (αh (k)) . According to Definition 5, item 2) the set Dm is a gyrotransversal. Let us study the transversal operation 〈E, (Dm) · , 1〉. We have: Dm (k1)Dm (k2) = (k1, α m k1 )(k2, α m k2 ) = = (k1α m k1 (k2) ,αm k1αm k1 (k2)) · (1, α(k1αm k1 (k2))−mkm 1 km 2 ) = = Dm(k1α m k1 (k2) ) · (1, αkm 1 (k1k2)−mkm 2 ), because of (k1α m k1 (k2) −m km 1 km 2 )−mkm 1 km 2 = ( km 1 k−1 2 k−1 1 k−m 1 )m km 1 km 2 = = ( km 1 k−1 2 k−1 1 k−m 1 ) ( km 1 k−1 2 k−1 1 k−m 1 ) · ... · ( km 1 k−1 2 k−1 1 k−m 1 ) ︸︷︷︸ m · km 1 km 2 = = km 1 ( k−1 2 k−1 1 )m km 2 = km 1 (k1k2) −m km 2 . It means that k1 (Dm) · k2 = k1α m k1 (k2) (33) and lk1,k2 = (1, αkm 1 (k1k2)−mkm 2 ). (34) Jo u rn al A lg eb ra D is cr et e M at h .E. Kuznetsov 79 Lemma 20. A left gyrogroup 〈E, (Dm) · , 1〉 is a group ⇔ the following identity (ab)m = bmam ∀a, b ∈ K/Z (K) . (35) is fulfilled in a factor-group K/Z (K). Proof. According to the formula (34), the left gyrogroup 〈E, (Dm) · , 1〉 is a group if and only if when αkm 1 (k1k2)−mkm 2 = id ∀k1, k2 ∈ K. It is equivalent to a fact that ∀a, b ∈ K/Z (K) it is true that am (ab)−m bm = 1, (ab)−m = a−mb−m, ( b−1a−1 )m = ( a−1 )m ( b−1 )m , (cd)m = dmcm ∀c, d ∈ K/Z (K) . Lemma 21. The left gyrogroup 〈E, (Dm) · , 1〉 is a gyrogroup ⇔ the follow- ing identity b2mam = ( a−mb am+1b )m ∀a, b ∈ K/Z (K) (36) is fulfilled in the factor-group K/Z(K). Proof. According to the formula (36), the left gyrogroup 〈E, (Dm) · , 1〉 is a gyrogroup if and only if when l a (Dm) · b,b = la,b, α (a (Dm) · b)m((a (Dm) · b)b)−mbm = αam(ab)−mbm , α(aamba−m)m(aamba−mb)−m bm = αam(ab)−mbm . It is equivalent to a fact that in the factor-group K/Z (K) it is true that ( am+1ba−m )m ( am+1ba−mb )−m bm = am (ab)−m bm, ( amaba−m ) · ( amaba−m ) · ... · ( amaba−m ) · ( am+1ba−mb )−m = am (ab)−m , (ab)m a−m ( am+1b a−mb )−m = (ab)−m , a−m ( am+1 b a−mb )−m = (ab)−2m , (ab)2m = ( am+1b a−mb )m am. Jo u rn al A lg eb ra D is cr et e M at h .80 Gyrogroups and left gyrogroups as transversals... Let us replace: c = ab, d = a−1; then c2m = ( d−m−1dc dmdc )m d−m, c2mdm = ( d−mc dm+1c )m ∀c, d ∈ K/Z (K) . Remark 6. If m = 1 then we obtain the results from [4]. References [1] Baer R.: Nets and groups. 1. Trans. Amer. Math. Soc., 46 (1939), 110-141. [2] Belousov V. D.: Foundations of quasigroup and loop theory (Russian), Moscow, “Nauka”, 1967. [3] T. Foguel: Groups, Transversals and loops, Comment. Math. Univ. Carolinae, 41, 2 (2000), 261-269. [4] T. Foguel, A.A. Ungar: Transversals, Loops and Gyrogroups, unpublished letter to the author from May 20, 1998. [5] T. Foguel, A.A. Ungar: Involutory decomposition of groups into twisted sub- groups and subgroups, J. Group Theory, 3 (2000), No. 1, 27-46. [6] T. Foguel, A.A. Ungar: Gyrogroups and the decompoziotion of groups into twisted subgroups and subgroups, Pasific J. Math., 197 (2001), 1-11. [7] Gwaramija A. A.: About one class of loops (Russian), Uchen. Zapiski, 375 (1978), 25-34. [8] Kargapolov M. I., Merzlyakov Yu. I., Foundations of group theory (Russian), Moscow, ”Nauka”, 1982. [9] Kiechle H.: Relatives of K-loops: Theory and examples, Comm. Math. Univ. Carolinae, 41, 2 (2000), No. 2, 201-324. [10] Kuznetsov E. A.: Transversals in groups. 1. Elementary properties, Quasi- groups and related systems, 1 (1994), 22-42. [11] Kuznetsov E.A.: Transversals in groups. 3. Semidirect product of a transversal operation and subgroup, Quasigroups and related systems, 8 (2001), 37-44. [12] Sabinin L. V.: On the equivalence of categories of loop and homogeneous spaces. Soviet Math. Dokl., 13 (1972), No. 4, 970-974. [13] Sabinin L. V., Mikheev O. I.: Quasigroups and differential geometry, Chap- ter XII in the book ”Quasigroups and loops: Theory and Applications”, Berlin, Helderman-Verlag, 1990, 357-430. [14] Abraham A. Ungar: Thomas rotation and the parametrization of the Lorentz transformation group, Found. Phys. Lett., 1 (1988), 57-89. [15] Ungar A.A.: Thomas Precession: Its Underlying Gyrogroup Axioms and Their Use in Hyperbolic Geometry and Relativistic Physics, Foundations of Physics, 27 (1997), No. 6, 881-951. Jo u rn al A lg eb ra D is cr et e M at h .E. Kuznetsov 81 Contact information E. Kuznetsov Institute of Mathematics and Computer Science, Academy of Sciences of Moldova, str., Academiei, 5, MD-2028, Chishinau, MOLDOVA E-Mail: ecuz@math.md, kuznet64@yahoo.com Received by the editors: 15.06.2003 and final form in 04.11.2003.

Gyrogroups and left gyrogroups as transversals of a special kind

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