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606_ies that it possesses. It suggests the pre x gyro that we extensively use to emphasize analogies

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     Involutory Decomposition of Groups Into Twisted Subgroups and Subgroups

     Tuval Foguel and Abraham A. Ungar Department of Mathematics North Dakota State University Fargo, ND 58105 USA e-mail:

    foguel@prairie.Nodak.edu e-mail: ungar@plains.Nodak.edu

     Suggested running title: Involutory Decomposition of Groups

     Typeset by AMS-TEX

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     Abstract. An involutory decomposition is a decomposition, due to an involution, of a group into a twisted subgroup and a subgroup. We study unexpected links between twisted subgroups and gyrogroups. Twisted subgroups arise in the study of problems in computational complexity. In contrast, gyrogroups are grouplike structures which rst arose in the study of Einstein's velocity addition in the special theory of relativity. Particularly, we show that every gyrogroup is a twisted subgroup and that, under general speci ed conditions, twisted subgroups are gyrocommutative gyrogroups. Moreover, we show that gyrogroups abound in group theory and that they possess rich structure.

     x1. Introduction

     Under general conditions, twisted subgroups are near subgroups 1]. Feder and Vardi 4] introduced the concept of a near subgroup of a nite group as a tool to study problems in computational complexity involving the class NP . Aschbacher provided a conceptual base for studying near subgroups demonstrating that near subgroups possess much structure, so that it seems likely that 1] "one can completely describe all near subgroups of nite groups] in some sense, using classi cation of the nite simple groups." Gyrogroups are essentially equivalent to gyrotransversals which are twisted subgroups, Theorem 3.8. Gyrogroups are special loops which share remarkable analogies with groups. The rst known gyrogroup structure is the relativistic gyrogroup (<3 ; ) that appeared in 1988 8], consisting of the unit ball <3 of the Euclidean 1 1 3-space <3 with Einstein's addition. The Einstein velocity addition of rela1 tivistically admissible velocities is a binary operation in the unit ball <3 = fx 2 1 <3 : jjxjj < 1g of the Euclidean 3-space <3 , where the vacuum speed of light is normalized to c = 1.

    Counterintuitively, the Einstein velocity addition is neither commutative nor associative. Is the progress from the common vector addition of velocities +, which is a group operation, to the Einstein velocity addition , which is not a group operation, associated with

    loss of mathematical regularity? It has been shown in 11] that the group structure that has been lost in the transition from the group (<3 ; +) to the nongroup groupoid (<3 ; ) is replaced by 1 a loop structure using a relativistic peculiar rotation called theThomas precession. Extending the Einstein relativistic groupoid (<3 ; ) with its Thomas precession by 1 abstraction, the gyrogroup notion emerges, where the abstract Thomas precession is called theThomas gyration. The Thomas gyration has its own life due to powerful properties that it possesses. It suggests the pre x gyro that we extensively use to emphasize analogies. Thus, for instance, gyrogroup operations are gyroassociative and, sometimes gyrocommutative, in full analogy with group operations which are associative and, sometimes commutative. Moreover, some commutative groups can be extended to vector spaces by the introduction of scalar multiplication and inner product and, in full analogy, some gyrocommutative gyrogroups can be extended to gyrovector spaces. Then, unexpectedly, gyrovector spaces provide the setting for hyperbolic geometry in full analogy with vector spaces, that provide the setting for Euclidean geometry 11,13]. Hence, the hyperbolic geometry of Bolyai-Lobachevski is in fact the gyro-Euclidean geometry. Like groups, there are nite and in nite gyrogroups some of which are gyrocommutative. We obtain particularly interesting results when the order of the group in which a gyrogroup resides as a twisted subgroup is odd. Unexpectedly, gyrogroups are twisted subgroups and, under speci ed general

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     conditions, twisted subgroups are gyrogroups. By demonstrating the rich structure of gyrogroups, and by demonstrating that gyrogroups and twisted subgroups are intimately related, we support Aschbacher's observation that 1] "near subgroups possess much structure." Furthermore, Aschbacher's hope to completely describe all near subgroups in some sense, using the classi cation of the nite simple groups, may result in the complete description of all nite gyrogroups as well. To show that gyrogroups abound in group theory we show in Theorem 5.1 that every odd order group that possesses an involutory automorphism contains a gyrocommutative gyrogroup. More speci cally, we nd that any involutory automorphism of an odd order group gives rise to a unique decomposition, called the involutory decomposition, that decomposes the group into the product of a twisted subgroup and a subgroup. We identify the twisted subgroup factor in the involutory decomposition as a gyrocommutative gyrogroup. Gyrogroups are, thus, everywhere in group theory, lying dormant waiting for their discovery. The gyrogroup de nition follows.

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     A groupoid 3] or a magma 2] is a nonempty set with a binary operation. An automorphism of the groupoid (S; ) is a bijection of S that respects

    the binary operation in S . The set of all automorphisms of (S; ) forms a group denoted by Aut(S; ). An important subcategory of the category of groupoids is the category of loops, which are de ned below. De nition 2.2. (Loops 3]) A loop is a magma (S; ) with an identity element in which each of the two equations a x = b and y a = b for the unknowns x and y possesses a unique solution (a x is the product of a and x in S . Subsequently we will omit the dot and write ax). Being nonassociative, the Einstein velocity addition of relativistically admissible velocities in the special theory of relativity is not a group operation. A gyrogroup is a special grouplike loop that has been abstracted from the groupoid of all relativistically admissible velocities with their Einstein's velocity addition and Thomas' precession 11]. The abstract Thomas precession is called the Thomas gyration, suggesting the pre x gyro that we extensively use to emphasize analogies. De nition 2.3. (Gyrogroups 11], Left Gyrogroups) The magma (G; ) is a gyrogroup if its binary operation satis es the following axioms. In G there is at least one element, 1, called a left identity, satisfying (G1) 1 a = a Left Identity for all a 2 G. There is an element 1 2 G satisfying axiom (G1) such that for each a in G there is an x in G, called a left inverse of a, satisfying (G2) x a = 1 Left Inverse Moreover, for any a; b; z 2 G there exists a unique element gyr a; b]z 2 G such that (G3) a (b z ) = (a b) gyr a; b]z Left Gyroassociative Law If gyr a; b] denotes the map gyr a; b] : G ! G given by z 7! gyr a; b]z then (G4) gyr a; b] 2 Aut(G; ) Gyroautomorphism and gyr a; b] is called the Thomas gyration, or the gyroautomorphism of G, generated by a; b 2 G. Finally, the gyroautomorphism gyr a; b] generated by any a; b 2 G satis es (G5) gyr a; b] = gyr a b; b] Left Loop Property A magma (G; ) is a left gyrogroup if it satis es axioms (G1)-(G4). De nition 2.4. (Gyrocommutative Gyrogroups) The gyrogroup (G; ) is gyrocommutative if for all a; b 2 G (G6) a b = gyr a; b](b a) Gyrocommutative Law

     x2. Gyrogroups, and Groups Containing Gyrogroups De nition 2.1. (Groupoids or Magmas, and Automorphism Groups of Groupoids)

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     As it is customary with groups, we use additive notation, , with gyrocommutative gyrogroups, and multiplicative notation, , with general gyrogroups. De nition 2.5. (Gyrations, Gyroautomorphisms, Gyroautomorphism Groups) The automorphisms gyr a; b] of a gyrogroup are called gyroautomorphisms. The action of the gyroautomorphism gyr a; b] on G is called a gyration. The set of all gyroautomorphisms of a gyrogroup (G; ) need not form a group. A gyroautomorphism group of (G; ) is any subgroup Auto (G; ) (not necessarily the smallest one) of Aut(G; ) containing all the gyroautomorphisms of (G; ). Properties of gyrogroups have been studied in 11] where, in particular, the following alternative, equivalent de nition of a gyrogroup is presented.

    Theorem 2.6 (Gyrogroups - an Alternative De nition 11]). A magma (G; ) is a gyrogroup if its binary operation satis es the following axioms and properties. In G there exists a unique element, 1, called the identity, satisfying

     (g1)

     1 a=a 1=a

     Identity Inverse

     for all a 2 G. For any a 2 G there exists in G a unique inverse, a?1 , satisfying (g2) a?1 a = a a?1 = 1 Moreover, if for any a; b 2 G the map gyr a; b] of G into itself is given by the equation gyr a; b]z = (a b)?1 (a (b z )) for all z 2 G, then the following hold for all a; b; c 2 G. (g3) gyr a; b] 2 Aut(G; ) (g4a) Gyroautomorphism Left gyroassociative Law Right gyroassociative Law Left Loop Property

     Right Loop Property Twisted subgroups prove useful as a tool to study problems in computational complexity 1]. We will see in this article that gyrogroups are intimately related to twisted subgroups, which are de ned below. De nition 2.7. (Twisted Subgroups 1]) A subset P of a group G is a twisted subgroup of G if (i) 1G 2 P , 1G being the identity element of G; and (ii) aPa P for all a 2 P . Every gyrogroup is a twisted subgroup of some speci ed group (Theorems 2.12 and 3.8), and some twisted subgroups are gyrogroups (Corollary 3.11). To expose the relationship between gyrogroups and twisted subgroups we will explore properties of gyrogroups in terms of groups that contain them as a subset. To enable us to study gyrogroups in terms of groups that may contain them we introduce the following de nitions and a theorem.

     a (b c) = (a b) gyr a; b]c (g4b) (a b) c = a (b gyr b; a]c) (g5a) gyr a; b] = gyr a b; b] (g5b) gyr a; b] = gyr a; b a]

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     De nition 2.8. (Transversals, the Transversal Operation, the Transversal Map, and Transversal Groupoids) A set B is a transversal in a group G (all transversals in this article are left transversals) of a subgroup H of G if every g 2 G can be written uniquely as g = bh where b 2 B and h 2 H . Let b1 ; b2 2 B be any two elements of B , and let

     (2.1)

     b1 b2 = (b1 b2 )h(b1 ; b2 )

     be the unique decomposition of the element b1 b2 2 G, where b1 b2 2 B and h(b1 ; b2 ) 2 H , determining (i) a binary operation, , in B , called the transversal operation of B induced by G, and (ii) a map h : B B ! H , called the transversal map. The element h(b1 ; b2) 2 H is called the element of H determined by the two elements b1 and b2 of its transversal B in G (its importance stems from the fact that it gives rise to gyrations in De nition 2.10 below. Gyrations, in turn, result from the abstraction of the Thomas precession of the special theory

    relativity into the Thomas gyration). A transversal groupoid (B; ) of H in G is a groupoid formed by a transversal B of H in G with its transversal operation . De nition 2.9. (Gyrotransversals, Gyrotransversal Groupoids, Gyro -Decompositions of Groups) A transversal groupoid (B; ) of a subgroup H in a group G is a gyrotransversal of H in G if (i) 1G 2 B , 1G being the identity element of G; (ii) B = B ?1 ; and

     (iii) B is normalized by H , H NG (B ), that is, hBh?1 B for all h 2 H . A gyrotransversal groupoid is a groupoid formed by a gyrotransversal with its transversal operation. The decomposition G = BH where H < G and where B is a transversal of H in G is a gyro-decomposition if B is a gyrotransversal of H in G. The gyro-decomposition G = BH is reduced if CH (B ) = f1Gg. Notation. In this paper we will use the notation bh = hbh?1 as in 5]. De nition 2.10. (Gyrations of a Gyrotransversal) Let B be a gyrotransversal of a subgroup H in a group G = BH , let b1 ; b2 2 B be any two elements of B , and let h(b1 ; b2 ) be the element of H determined by b1 and b2 , h being the transversal map h : B B ! H , Def. 2.8. Then the gyration gyr b1 ; b2 ] of B generated by b1 and b2 is the map of B into itself given by

     gyr b1 ; b2 ] =

     where

     h h

     h(b1 ;b2 )

     , h 2 H , denotes conjugation by h, that is, for any h 2 H and b 2 B (b) = bh = hbh?1

     It follows from De nition 2.10 that a gyration of B generated by b1 ; b2 2 B is given in terms of its e ects on x 2 B by the equation (2.2a)

     gyr b1 ; b2 ]x = h(b1 ; b2 )x(h(b1 ; b2 ))?1

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     or, equivalently, by the equation (2.2b)

     gyr b1 ; b2 ]x = xh(b1 ;b2 )

     The conjugation operations h , h 2 H , are bijections of B since B is normalized by H . Hence, in particular, the gyrations gyr b1 ; b2 ] are bijections of B for all b1 ; b2 2 B . Moreover, the gyrations of a gyrotransversal B are automorphisms of the gyrotransversal groupoid (B; ) as shown in the following Theorem 2.11. Let (B; ) be a gyrotransversal groupoid of a subgroup H in a group G. Then, for any b1; b2 2 B

     gyr b1 ; b2 ] 2 Aut(B; )

     Proof. Since, by Eq. (2.2b), gyr a; b]x = xh(a;b) for all x 2 B , we have to show that

     (x y)h(a;b) = xh(a;b) yh(a;b) for all a; b; x; y 2 B . More generally, however, we will verify the desired identity for any k 2 H regardless

    of whether or not k possesses the form k = h(a; b). We will thus show that (x y)k = xk yk for any k 2 H . Clearly, we have in G (2.3) (xy)k = xk yk Employing the unique decomposition G = BH , Eq. (2.1), for both sides of (2.3) we have (2.4) on one hand, and (2.5) (xy)k = ((x y)h(x; y))k = (x y)k h(x; y)k

     xk yk = (xk yk )h(xk ; yk )

     on the other hand. It follows from (2.3) - (2.5) and from the uniqueness of the decomposition G = BH that (2.6) and, by the way, (2.7) Eq. (2.6) completes the proof. (x y)k = xk yk

     h(xk ; yk ) = h(x; y)k

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     Theorem 2.12. (Representation Theorem for Gyrogroups) If (B; ) is a leftgyrogroup and H a gyroautomophism group of (B; ), then there is a group G in which H is a subgroup of G and (B; ) is a gyrotransversal groupoid of H such that for each h 2 H and x 2 B , h(x) = xh . Proof. This result is proved in section 4 of 11] for gyrogroups, but the proof does not use the left loop property so that it is valid for left gyrogroups. Theorem 2.13. Any gyrotransversal B of a subgroup H in G is a left gyrogroup Proof. We have to show that (B; ) satis es axiomas (G1)-(G4). Axiom (G4) is veri ed in Theorem 2.11. It therefore remains to establish the validity of axioms (G1)-(G3). Given b 2 B we get

     b = 1b = (1 b)h(1; b) and 1 = b?1b = (b?1 b)h(b?1 ; b)

     Hence,(G1) and (G2) are veri ed from the uniqueness of the decomposition. for all a; b; c 2 B we clearly have in G (2.8) (2.9) (ab)c = a(bc) (ab)c = (a b)h(a; b)c = (a b)gyr a; b]ch(a; b) = ((a b) gyr a; b]c)h(a b; gyr a; b]c)h(a; b) Employing the uniqueness of the decomposition for both sides of (2.8) we have

     on one hand, and (2.10)

     a(bc) = a(b c)h(b; c) = (a (b c))h(a; b c)h(b; c)

     on the oher hand. It follows from (2.8)-(2.10) and from the uniqueness of the decomposition that (a b)gyr a; b]c = a (b c) thus verifying (G3).

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     We will show in this section that a gyrotransversal groupoid in a group is a gyrogroup if and only if the gyrotransversal is a twisted subgroup of G when the group G containing the twisted subgroup is reduced in the sense of De nition 3.4 below. In the following Lemma, CH (B ) denotes the centralizer of B in H < G. Lemma 3.1. Let G = BH be a group where H < G. If H NG(B) then CH (B) / G. Proof. Let A = hB i. Since G = BH , we have G = AH and since H acts on B , H acts on A, so CH (B ) = CH (A) E H . Finally A acts on CH (A), so G = AH NG(CH (A)). Lemma 3.2. Let (B; ) be the gyrotransversal groupoid of a subgroup H of a group G, G = BH . Then (B; ) is the gyrotransversal groupoid of the subgroup HB = H=CH (B ) in the group GB = G=CH (B ). Proof. The permutation

    representation of G on G=H is equivalent to the representation of G on B by conjugation. The group GB is just the image of G under this representation. De nition 3.3. (Transversal Enveloping Pairs) Let B be a transversal of a subgroup H in a group G. We say that G (H ) is an enveloping group (subgroup) of the transversal B . Furthermore, we say that (G; H ) is an enveloping pair of the transversal B and of the transversal groupoid (B; ). De nition 3.4. (Reduced Enveloping Pairs of Gyrotransversals) Let B be a gyrotransversal with an enveloping pair (G; H ). The corresponding reduced enveloping pair of the gyrotransversal B and of the gyrotransversal groupoid (B; ) is the pair (GB ; HB ) = (G=CH (B ); H=CH (B )) It follows from Lemma 3.2 that a reduced enveloping pair of a gyrotransversal groupoid is an enveloping pair of the gyrotransversal groupoid. We clearly have the following Lemma, which exposes the importance of reducing enveloping pairs. Lemma 3.5. If the enveloping pair (G; H ) of the gyrotransversal B of H in G is reduced, that is, (G; H ) = (GB ; HB ), then CH (B ) = f1H g is the trivial group consisting of the identity element of H . Theorem 3.6. Let B be a gyrotransversal with a reduced enveloping pair (G; H ). Then the map h 7! h is a bijection of h(B B ) with the set of all gyrations of B . Proof. By de nition, the gyrations gyr b1 ; b2] of B , b1 ; b2 2 B correspond to elements of h(B B ) H by the relation

     x3. Gyrotransversals are Twisted Subgroups

     gyr b1 ; b2 ] =

     where

     h(b1 ;b2 )

     h h is the inner automorphism of B given by h b = b for all b 2 B and h 2 H . And the map : h 7! h is an injective group homomorphism from H into the symmetric group on B as ker( ) = CH (B ) = f1H g, so that its restriction to h(B B ) is injective.

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     It follows from Lemma 3.2 and De nition 3.3 that in the study of gyrotransversal groupoids on their own merits, rather than on merits of the group where they reside, one may assume without loss of generality that any gyrotransversal under consideration resides in one of its reduced enveloping groups. This results in the advantage of having a bijective correspondence between gyrations and transversal maps. Speci cally, let (B; ) be a gyrotransversal groupoid with a reduced enveloping pair (G; H ). Then, there exists a bijective correspondence between the gyrations gyr b1 ; b2 ] 2 Aut(B; ) of the gyrotransversal B and the elements h(b1 ; b2) 2 H of the image in H of the transversal map h. As an example, we present the reduced enveloping pair of Einstein's gyrogroup (<n ; E ), where <n = fv2<n : jjvjj < cg is c c the open c-ball of the Euclidean n-space <n , and where E is the Einstein addition, de ned in 11] (No explicit presentation of E is needed in Example 3.7).

    Example 3.7. The Lorentz group, parametrized by a velocity and an orientation parameter, is a group of pairs (3.1a) L = f(v; V ) : v 2 <n ; V 2 SO(n)g c with group operation given by (3.1b) (u; U )(v; V ) = (u E U v; gyr u; U v]UV ) where E is the Einstein velocity addition 11] and where SO(n) is the special orthogonal group. The Lorentz group L and the orthagonal group SO(n) constitute a reduced enveloping pair, (L; SO(n)), of the Einstein gyrotransversal (<n ; E ). c We are now in a position to state the conditions under which twisted subgroups and gyrogroups are equivalent. Theorem 3.8. A gyrotransversal groupoid (P; ) with a reduced enveloping pair (G; H ) is a gyrogroup if and only if P is a twisted subgroup of G, and h(a; b) = h?1 (b; a) (note from the proof that h(a; b) = h?1(b; a) is satis ed in the gyrocommutative case). Proof. Let us assume that P is a twisted subgroup. Then, aba 2 P for any a; b 2 P . We wish to show that the gyrotransversal groupoid (P; ) of H in G is a gyrogroup. Clearly, bb = b1Gb 2 P . Hence, abba 2 P . But, in G for the gyrocommutative case, abba = (ab)(ba) = (a b)h(a; b)(b a)h(b; a) = (a b)(b a)h(a;b) h(a; b)h(b; a) implying h(a; b)h(b; a) = 1H Hence, h?1 (b; a) = h(a; b) Similarly, since aba 2 P we have in G aba = a(ba) = a(b a)h(b; a) = a (b a)h(a; b a)h(b; a)

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     implying so that

     h(a; b a)h(b; a) = 1H

     h(a; b a) = h?1 (b; a) = h(a; b) thus obtaining the right loop property h(a; b a) = h(a; b). Inverting by means of h?1 (a; b) = h(b; a) we obtain the desired left loop property for h, h(b a; a) = h(b; a) for all a; b 2 P . Since gyr a; b] = h(a;b) we have gyr a b; b] = gyr a; b]

     for all a; b 2 P . Hence, the gyrotransversal groupoid (P; ) possesses the left loop property. By Theorem 2.13 the gyrotransversal groupoid (P; ) is therefore a gyrogroup. Conversely, we now assume that (P; ) is a gyrogroup. Let a; b 2 P be any two elements of P . We will show that the composition aba in G is an element of P . Gyrations gyr a; b] in P are in bijective correspondence with elements h(a; b) 2 H of the image h(B B ) of h in H , by Theorem 3.6. Hence, the (left and) right loop property for gyr a; b] is valid for h(a; b) as well, that is, gyr a; b] = gyr a; b a], implying h(a; b) = h(a; b a). Similarly, the identity gyr?1 a; b] = gyr b; a] implies h?1 (a; b) = h(b; a). Following these properties of h we have in G

     aba = a(ba) = a(b a)h(b; a) = a (b a)h(a; b a)h(b; a) = a (b a)h(a; b)h(b; a) = a (b a) 2 P

     for all a; b 2 P . Hence, P is a twisted subgroup of G, thus completing the proof. Corollary 3.9. Let G = AH , H < G, be a gyro-decomposition of G, Def. 2.9. If A is a twisted subgroup of G then A is a gyrogroup. Proof. The proof follows from the rst part of the proof of Theorem 3.8. Corollary 3.9 suggests the following de nition. De nition 3.10.

    (Gyro-Twisted Subgroups) A twisted subgroup P in a group G is a gyro-twisted subgroup if P is a gyrotransversal of some subgroup H < G in G. Following De nition 3.10, Corollary 3.9 can now be stated as Corollary 3.11. Any gyro-twisted subgroup P in a group G is a gyrogroup (P; ), whose gyrogroup operation is the transversal operation of P induced by G. Example 3.12. The most general Mobius transformation of the complex unit disc D = fz : jz j < 1g in the complex z -plane 12], 0 z 7! ei 1z++ zz = ei (z0 z ) z0

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     de nes the Mobius addition in the disc, allowing the Mobius transformation of the disc to be viewed as a Mobius left translation z 7! z0 z followed by a rotation. Here 2 < is a real number, z0 2 D, and z0 is the complex conjugate of z0 . The Mobius addition of two real numbers in the disc specializes to the Einstein velocity addition of parallel velocities in the special theory of relativity. A left Mobius translation is also called a left gyrotranslation 11]. Left gyrotranslations occur frequently in hyperbolic geometry 12], and are sometimes called hyperbolic pure translations. The Mobius transformations of the disc D form a group, M . The rotations z 7! ei z of the disc about its center form a subgroup R of M . In contrast, the left gyrotranslations z 7! z0 z of the disc do not form a subgroup of M . They do, however, form a twisted subgroup T of M . Furthermore, T is a transversal of R in M , T ?1 = T and T is normalized by R in M . Hence, the twisted subgroup T of M is a gyro-twisted subgroup. As such, according to Corollary 3.11, the groupoid (T; ) formed by the gyro-twisted subgroup T of R in M with its transversal operation is a gyrogroup. This gyrogroup is studied in 10].

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     x4. Involutory Decompositions and Gyrocommutative Gyrogroups De nition 4.1. (Involutory automorphisms) An automorphism of a group G is

     involutory if it equals its inverse automorphism. The main result of this article is presented in the following theorem, demonstrating that gyrocommutative gyrogroups are associated with involutory automorphisms that groups in which they reside as subsets must possess. Theorem 4.2. Let G = AH be a reduced gyro-decomposition of a group G, H < G. If A is a gyrocommutative gyrogroup, then there exists an involutory automorphism 2 Aut(G) such that (h) = h for all h 2 H , and (a) = a?1 for all a 2 A. Proof. We de ne : G ! G by (g) = a?1 h for g = ah 2 G, a 2 A and h 2 H . Clearly, 2 = 1, and is bijective. It remains to show that is a homomorphism. For this we need to use Theorem 5.5 of 11] according to which h(a?1 ; b?1 ) = h(a; b) for a; b 2 A in any Gyrogroup A, and Theorem 5.9 of 11] according to which (a b)?1 = a?1 b?1 for a; b 2 A if and only if the gyrogroup A is Gyrocommutative.

    Let g1 = a1 h1 and g2 = a2 h2 be any two elements of G. On one hand (g1 ) (g2 ) = (a1 h1 ) (a2 h2 ) = a?1 h1 a?1 h2 1 2 ? = a1 1 (a?1 )h1 h1 h2 2 ? ? = a?1 (a2 1 )h1 h(a1 1 ; (a?1 )h1 )h1 h2 1 2 ?1 (ah1 )?1 h(a?1 ; (ah1 )?1 )h1 h2 = a1 2 1 2 ?1 (ah1 )?1 h(a1 ; ah1 )h1 h2 = a1 2 2 and, on the other hand, (g1 g2 ) = (a1 h1 a2 h2 ) = (a1 ah1 h1 h2 ) 2 = (a1 ah1 h(a1 ; ah1 )h1 h2 ) 2 2 = (a1 ah1 )?1 h(a1 ; ah1 )h1 h2 2 2 ? = a1 1 (ah1 )?1 h(a1 ; ah1 )h1 h2 2 2 Hence, (g1 g2 ) = (g1 ) (g2 )

     as desired. Theorem 4.2 suggests the following two de nitions.

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     De nition 4.3. (Inverters and Stabilizers of Automorphisms in Groups) For any automorphism 2 Aut(G) of a group G let the subset K ( ) and the subgroup C ( )

     be given by

     K ( ) = fg 2 Gj (g) = g?1 g C ( ) = fg 2 Gj (g) = gg

     The subset K ( ) of G is called the inverter of in G, and the subgroup C ( ) of G is called the stabilizer of in G. It follows from Theorem 4.2 that if G = AH , H < G, is a gyro-decomposition of a group G, and if A is a gyrocommutative gyrogroup, then there exists an automorphism 2 Aut(G) such that H C ( ), A K ( ), and G = K ( )C ( ). This suggests the following de nition and theorem. De nition 4.4. (Involutory Decompositions) A gyro-decomposition (Def. 2.9) G = AH of a group G is involutory, with respect to an involutory automorphism , if there exists an involutory automorphism 2 Aut(G) such that A K ( ) and H C ( ). Theorem 4.5. Let G = AH be a gyro-decomposition of a group G where (i) A is a twisted subgroup of G and (ii) H is a subgroup of G. Then, the decomposition G = AH is involutory if and only if the transversal groupoid (A; ) is a gyrocommutative gyrogroup. Proof. Let G = AH be a gyro-decomposition of G where A is a twisted subgroup of G and H is a subgroup of G. To verify that involutory decomposition implies gyrocommutivity we assume that the decomposition is involutory. Since the decomposition is a gyro-decomposition, Def. 2.9, A is a gyrotransversal of H in G. Hence, by Def. 3.10, A is a gyro-twisted subgroup of G. Hence, by Corollary 3.11, (A; ) is a gyrogroup whose operation is the transversal operation of A induced by G. It remains to verify that the gyrogroup (A; ) is gyrocommutative. Let a; b 2 A. On one hand (ab) = ((a b)h(a; b)) = (a b)?1 h(a; b) and on the other hand, (ab) = (a) (b) = a?1 b?1 = (a?1 b?1 )h(a?1 ; b?1 ) Hence, by the unique decomposition (a b)?1 = a?1 b?1 so that by Theorem 5.9 in 11], the gyrogroup A is gyrocommutative. Conversely, let G = AH be a gyro-decomposition of G as above. We now assume that the transversal groupoid (A; ) is a gyrocommutative gyrogroup, and wish to

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     show that the decomposition is involutory. This follows immediately

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