Gyrogroup actions: A generalization of group actions

0.17 MB·

by  Teerapong Suksumran

Gyrogroup actions:
6 A generalization of group actions∗,†
1
0
2
Teerapong Suksumran‡

b
e DepartmentofMathematics
F
NorthDakotaStateUniversity

4 Fargo, ND 58105,USA

] [email protected]
R
G
.
h
t Abstract
a
m Thisarticleexploresthenovelnotionofgyrogroupactions,whichisanatural
generalizationoftheusualnotionofgroupactions.Asafirststeptowardthestudy
[
of gyrogroup actions from the algebraic viewpoint, we prove three well-known

2 theoremsingrouptheoryforgyrogroups:theorbit-stabilizertheorem,theorbitde-
v compositiontheorem,andtheBurnsidelemma(ortheCauchy-Frobeniuslemma).
8 Wethenprovethatunder acertaincondition, agyrogroup Gactstransitivelyon
9
thesetG/H ofleftcosetsofasubgyrogroupH inGinanaturalway. Fromthis
4
we prove the structure theorem that every transitive action of a gyrogroup can
6
berealizedasagyrogroup actionbyleftgyroaddition. Wealsoexhibitconcrete
0
examplesofgyrogroupactionsfromtheMo¨biusandEinsteingyrogroups.
.
1
0 Keywords. permutation representation, gyrogroup action, orbit-stabilizer theorem,
6 Burnsidelemma,gyrogroup,leftgyroaddition.
1
2010MSC.Primary20C99;Secondary05A05,05A18,20B30,20N05.
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v
i
X
1 Introduction
r
a
The methodof groupaction is consideredas an importantand a powerfultoolin
mathematics. It is used to unravel many of mathematical structures. In fact, when
onestructureactsonanotherstructure,abetterunderstandingisobtainedonboth.The
Sylowtheorems,forinstance,resultfromtheactionofagrouponitselfbyconjugation.
∗ThisisthefinalversionofthemanuscriptappearedinJ.Algebra454(2016),70–91. Thepublished
versionofthearticleisaccessibleviadoi:10.1016/j.jalgebra.2015.12.033.
†PartofthisworkhasbeenpresentedattheJointMathematics Meetings (JMM2016), Seattle, WA,
USA,January6–9,2016.
‡TheauthorwasfinanciallysupportedbyInstituteforPromotionofTeachingScienceandTechnology
(IPST),Thailand,throughDevelopmentandPromotionofScienceandTechnologyTalentsProject(DPST).
(cid:13)c 2016 Author. This manuscript version is made available under the CC-BY-NC-ND 4.0 license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
T. Suksumran Gyrogroup actions 2
The algebraic structure of a finite group in turn is revealed by the use of the Sylow
theorems.Further,manywide-applicationtheoremsingrouptheoryandcombinatorics
suchastheorbit-stabilizertheorem,theclassequation,andtheBurnsidelemma(also
called the Cauchy-Frobenius lemma) are byproducts of group actions. In addition,
groupactionsareusedtoproveversionsoftheLagrangetheorem,theSylowtheorems,
andtheHalltheoremsforBruckloopsin[2].
Therearesomeattemptstostudypermutationrepresentationsofquasigroupsand
loops—algebraicstructuresthatareageneralizationofgroups. Forinstance,Jonathan
Smithhasintensivelystudiedquasigroupandlooprepresentationsinaseriesofpapers
[9–11]. Also, the study of sharply transitive sets in quasigroupactions can be found
in[7].
Gyrogroupsare a suitable generalizationof groups, arising from the study of the
parametrization of the Lorentz transformation group by Abraham Ungar [16]. The
origin of a gyrogroupis described in [19] and references therein. Gyrogroupsshare
remarkable analogies with groups. In fact, every group forms a gyrogroup under
the same operation. Many of classical theorems in group theory continue to hold
for gyrogroups, including the Lagrange theorem [13], the fundamentalisomorphism
theorems[15],andtheCayleytheorem[15]. Further,harmonicanalysiscanbestudied
in the framework of gyrogroups[4,5], using the gyroassociative law in place of the
associativelaw.
Inthisarticle,wecontinuetoprovesomewell-knowntheoremsingrouptheoryfor
gyrogroups,includingtheorbit-stabilizertheorem,theBurnsidelemma,andtheorbit
decomposition theorem. These theorems are proved by techniques similar to those
used in group theory, where gyroautomorphisms play the fundamental role and the
associativelawisreplacedbythegyroassociativelaw. Werecovertheclassicalresults
if a gyrogroupis degeneratein the sense thatits gyroautomorphismsare the identity
automorphism. It is worth pointingoutthatthe resultsinvolvinggyrogroupsmay be
recastintheframeworkofleftBolloopswiththepropertythatleftinnermappingare
automorphisms.
The structure of the article is organized as follows. In Section 2, we review the
basictheoryofgyrogroups.InSection3,westudygyrogroupactionsor,equivalently,
permutationrepresentationsofagyrogroup.InSection4,weprovethatunderacertain
condition, a gyrogroup G acts transitively on the coset space G/H of left cosets of
a subgyrogroup H in G by left gyroaddition. This results in the fundamental iso-
morphismtheoremforgyrogroupactions. InSection5,weexhibitconcreteexamples
oftransitivegyrogroupactionsfromtheMo¨biusandEinsteingyrogroups.
2 Gyrogroups
We summarizebasicpropertiesofgyrogroupsforreference. Muchofthissection
canbefoundin[13,15,19]. Thereaderfamiliarwithgyrogrouptheorymayskipthis
section.
A pair (G,⊕) consisting of a nonempty set G and a binary operation ⊕ on G is
called a groupoid. The group of automorphismsof a groupoid(G,⊕) is denoted by
Aut(G,⊕). Ungar[19]formulatestheformaldefinitionofagyrogroupasfollows.
T. Suksumran Gyrogroup actions 3
Definition2.1(Gyrogroups). Agroupoid(G,⊕)isagyrogroupifitsbinaryoperation
satisfiesthefollowingaxioms.
(G1) Thereisanelement0∈Gsuchthat0⊕a=aforalla∈G.
(G2) Foreacha∈G,thereisanelementb∈Gsuchthatb⊕a=0.
(G3) Foralla,b∈G,thereisanautomorphismgyr[a,b]∈Aut(G,⊕)suchthat
a⊕(b⊕c)=(a⊕b)⊕gyr[a,b]c (leftgyroassociativelaw)
forallc∈G.
(G4) Foralla,b∈G,gyr[a⊕b,b]=gyr[a,b]. (leftloopproperty)
We remarkthatthe axiomsinDefinition2.1implythe rightcounterparts. Inpar-
ticular,anygyrogrouphasauniquetwo-sidedidentity0,andanelementaofthegyro-
group has a unique two-sided inverse ⊖a. Let G be a gyrogroup. For a,b∈G, the
mapgyr[a,b]iscalledthegyroautomorphismgeneratedbyaandb. ByTheorem2.10
of[19],thegyroautomorphismsarecompletelydeterminedbythegyratoridentity
gyr[a,b]c=⊖(a⊕b)⊕(a⊕(b⊕c)) (1)
foralla,b,c∈G. Notethateverygroupformsagyrogroupunderthesameoperation
bydefiningthegyroautomorphismstobetheidentityautomorphism,buttheconverse
isnotingeneraltrue. Fromthispointofview,gyrogroupssuitablygeneralizegroups.
Recallthatthecoaddition,⊞,ofagyrogroupGisdefinedbytheequation
a⊞b=a⊕gyr[a,⊖b]b (2)
fora,b∈G. Seta⊟b=a⊞(⊖b)fora,b∈G. ByTheorem2.22of[19],thefollowing
cancellationlawsholdingyrogroups.
Theorem2.2([19]). LetGbeagyrogroup.Foralla,b,c∈G,
(1) a⊕b=a⊕cimpliesb=c; (generalleftcancellationlaw)
(2) ⊖a⊕(a⊕b)=b; (leftcancellationlaw)
(3) (b⊖a)⊞a=b; (rightcancellationlawI)
(4) (b⊟a)⊕a=b. (rightcancellationlawII)
Subgyrogroups,gyrogrouphomomorphisms,andquotientgyrogroupsarestudied
in[13,15]. AnattempttoproveagyrogroupversionoftheLagrangetheoremleadsto
thenotionofL-subgyrogroups[15]. AsubgyrogroupH ofagyrogroupGiscalledan
L-subgyrogroupifgyr[a,h](H)=H foralla∈Gandh∈H. L-subgyrogroupsbehave
well in the sense that they partition G into left cosets of equal size. More precisely,
if H isan L-subgyrogroupof G, then H anda⊕H :={a⊕h: h∈H} havethesame
cardinalityandthecosetspaceG/H:={a⊕H: a∈G}formsadisjointpartitionofG.
InthecasewhereGisafinitegyrogroup,weobtainthefamiliarindexformula
|G|=[G: H]|H|. (3)
T. Suksumran Gyrogroup actions 4
Here, [G: H] denotes the index of H in G, which is defined as the cardinality of the
cosetspace G/H. We will see in Section4 thatcertain L-subgyrogroupsgive rise to
gyrogroupactionsbyleftgyroaddition.
Foranynon-L-subgyrogroupK ofagyrogroupG, itisnolongertruethattheleft
cosetsofKpartitionG. Moreover,theindexformula|G|=[G: K]|K|isnotingeneral
true. Nevertheless,theorderofanysubgyrogroupofafinitegyrogroupGdividesthe
orderofG,see[13,Theorem5.7].
3 Gyrogroup actions
Throughouttheremainderofthearticle,X isa(finiteorinfinite)nonemptysetand
Gisa(finiteorinfinite)gyrogroupunlessotherwisestated.
3.1 Definition
Definition3.1(Gyrogroupactions). AmapfromG×X toX,written(a,x)7→a·x,is
a(gyrogroup)actionofGonX ifthefollowingconditionshold:
(1) 0·x=xforallx∈X,and
(2) a·(b·x)=(a⊕b)·xforalla,b∈G,x∈X.
Inthiscase,X issaidtobeaG-setandGissaidtoactonX.
Note that if G is a gyrogroupwith trivial gyroautomorphisms,then the notion of
gyrogroup actions specializes to the usual notion of group actions. In other words,
gyrogroupactionsnaturallygeneralizegroupactions. We willpresentsomeconcrete
examplesofgyrogroupsthatsatisfytheaxiomsofagyrogroupactioninSection5.
LetSym(X)denotethegroupofpermutationsofX. Sinceeverygroupisagyro-
groupunderthesameoperation,Sym(X),togetherwithfunctioncomposition◦,forms
agyrogroupwithtrivialgyroautomorphisms.Thus,itmakessensetospeakofagyro-
group homomorphismfrom G to Sym(X). Recall that a map j : G→Sym(X) is a
gyrogrouphomomorphismif
j (a⊕b)=j (a)◦j (b) (4)
foralla,b∈G.
ThefollowingtheoremassertsthatanygyrogroupactionofGonX inducesagyro-
grouphomomorphismfromGtoSym(X). LetX beaG-set.Foreacha∈G,defines
a
by
s (x)=a·x, x∈X. (5)
a
Definej˙ bytheequation
j˙(a)=s , a∈G. (6)
a
T. Suksumran Gyrogroup actions 5
Theorem3.2. LetX beaG-set.
(1) Foreacha∈G,s definedby(5)isapermutationofX.
a
(2) Themapj˙ definedby(6)isagyrogrouphomomorphismfromGtoSym(X). Its
kerneliskerj˙ ={a∈G: a·x=x forallx∈X}.
Proof. Thetheoremfollowsdirectlyfromtheaxiomsofagyrogroupaction. Notethat
a∈kerj˙ ifandonlyifs =id ifandonlyifa·x=xforallx∈X.
a X
AccordingtoTheorem3.2, j˙ iscalledthegyrogrouphomomorphismaffordedby
agyrogroupactionofGonX ortheassociatedpermutationrepresentationofG. The
processofturningagyrogroupactionintoapermutationrepresentationisreversiblein
thesenseofthefollowingtheorem.
Theorem3.3. Letj : G→Sym(X)beagyrogrouphomomorphism.Themap·defined
bytheequation
a·x=j (a)(x), a∈G,x∈X, (7)
isagyrogroupactionofGonX. Furthermore,j˙ =j .
Proof. Clearly, 0·x=j (0)(x)=id (x)=x for all x∈X. For all a, b∈G, x∈X,
X
we have a·(b·x)=(j (a)◦j (b))(x)=j (a⊕b)(x)=(a⊕b)·x. By construction,
j˙(a)(x)=a·x=j (a)(x)forallx∈X,a∈G. Hence,j˙ =j .
Theorems3.2and3.3togetherimplythatthestudyofgyrogroupactionsofGonX
isequivalenttothestudyofgyrogrouphomomorphismsfromGtoSym(X).
3.2 Orbits andstabilizers
Inthissection,weprovegyrogroupversionsofthreewell-knowntheoremsingroup
theoryandcombinatorics:
• theorbit-stabilizertheorem;
• theorbitdecompositiontheorem;
• theBurnsidelemma.
WewillseeshortlythatstabilizersubgyrogroupsofGhaveniceproperties.Forin-
stance,theyshareremarkablepropertieswithstabilizersubgroups;areL-subgyrogroups
andhencepartitionGintoleftcosets;andareinvariantunderthegyroautomorphisms
ofG. Amongotherthings,theyleadtotheorbit-stabilizertheoremforgyrogroups.
LetX beaG-set. Definearelation∼onX bythecondition
x∼y ⇔ y=a·xforsomea∈G. (8)
Theorem3.4. Therelation∼definedby(8)isanequivalencerelationonX.
T. Suksumran Gyrogroup actions 6
Proof. Letx,y,z∈X.
Reflexive.Since0·x=x,wehavex∼x.
Symmetric. Supposethatx∼y. Theny=a·xforsomea∈G. Since(⊖a)·y=x,
wehavey∼x.
Transitive. Suppose that x∼y and y∼z. Then y=a·x and z=b·y for some
a,b∈G. Sincez=b·(a·x)=(b⊕a)·x,wehavex∼z.
Let x∈X. The equivalenceclass of x determinedby the relation ∼ is called the
orbitofxandisdenotedbyorbx,thatis,orbx={y∈X: y∼x}. Itisstraightforward
tocheckthat
orbx={a·x: a∈G} (9)
forallx∈X. ThestabilizerofxinG,denotedbystabx,isdefinedas
stabx={a∈G: a·x=x}. (10)
Proposition3.5. LetX beaG-set.
(1) Foreachx∈X,stabxisasubgyrogroupofG.
(2) kerj˙ = stabx.
x∈X
\
Proof. Clearly, 0 ∈stabx. Let a, b ∈stabx. Since (a⊕b)·x = a·(b·x) =x, we
have a⊕b∈stabx. Also, (⊖a)·x=(⊖a)·(a·x)=x. Hence, ⊖a∈stabx. By the
subgyrogroupcriterion[15,Proposition14],stabx6G. Item(2)followsdirectlyfrom
Theorem3.2(2).
Proposition3.6. LetX beaG-set. Foralla,b∈G,x∈X,
gyr[a,b](stabx)⊆stabx.
Inparticular,ifc∈Gandc·x=x,then(gyr[a,b]c)·x=x.
Proof. Letc∈stabx. Accordingtothegyratoridentity(1),wecompute
(gyr[a,b]c)·x=[⊖(a⊕b)⊕(a⊕(b⊕c))]·x
=⊖(a⊕b)·[a·(b·(c·x))]
=⊖(a⊕b)·[a·(b·x)]
=⊖(a⊕b)·[(a⊕b)·x]
=x.
Hence,gyr[a,b]c∈stabx,whichcompletestheproof.
Corollary3.7. LetX beaG-set. Foralla,b∈G,x∈X,
(1) gyr[a,b](stabx)=stabx,and
(2) stabxisanL-subgyrogroupofG.
T. Suksumran Gyrogroup actions 7
Proof. Item (1) follows directly from Proposition 6 of [15]. By item (1), stabx is
invariantunderthegyroautomorphismsofG. Hence,stabx6 G.
L
Lemma3.8. LetX beaG-set. Foralla,b∈G,x∈X,thefollowingareequivalent:
(1) a·x=b·x;
(2) (⊖b⊕a)·x=x;
(3) a⊕stabx=b⊕stabx.
Proof. Theproofoftheequivalence(1)⇔(2)isstraightforward,usingtheaxiomsof
agyrogroupaction.ByCorollary3.7,stabx6 G. Itfollowsthat
L
(⊖b⊕a)·x=x ⇔ ⊖b⊕a∈stabx ⇔ a⊕stabx=b⊕stabx.
Thisprovestheequivalence(2)⇔(3).
Wearenowinapositiontostateagyrogroupversionoftheorbit-stabilizertheorem.
Theorem3.9(Theorbit-stabilizertheorem). LetGbeagyrogroupactingonasetX.
Foreachx∈X,thereexistsabijectionfromtheorbitofxtothecosetspaceG/stabx.
Inparticular,ifGisafinitegyrogroup,then
|G|=|orbx||stabx|. (11)
Proof. Letq bethemapdefinedonorbxby
q (a·x)=a⊕stabx, a∈G.
By Lemma 3.8, q is well defined and injective. That q is surjective is clear. Since
stabxisanL-subgyrogroupofG,(3)implies|G|=[G: stabx]|stabx|. Because
[G: stabx]=|G/stabx|=|orbx|,
weobtain|G|=|orbx||stabx|.
LetX beaG-set. ThesetoffixedpointsofX,denotedbyFix(X),isdefinedas
Fix(X)={x∈X: a·x=xforalla∈G}. (12)
From(9)onefindsthatx∈Fix(X)ifandonlyiforbx={x}. Thefollowingtheorem
can be regarded as a generalization of the class equation familiar from finite group
theory.
Theorem3.10(Theorbitdecompositiontheorem). LetGbeagyrogroupactingon
afinitesetX. Letx ,x ,...,x berepresentativesforthedistinctnonsingletonorbits
1 2 n
inX. Then
n
(cid:229)
|X|=|Fix(X)|+ [G: stabx]. (13)
i
i=1
T. Suksumran Gyrogroup actions 8
Proof. Since{orbx: x∈X}formsadisjointpartitionofX,itfollowsfromtheorbit-
stabilizertheoremthat
n
|X|= orbx + orbx
i
(cid:12)x∈F[ix(X) (cid:12) (cid:12)i[=1 (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:229) n(cid:12) (cid:12) (cid:12)
=|Fix(X)|+ |orbx|
i
i=1
n
(cid:229)
=|Fix(X)|+ [G: stabx].
i
i=1
IfGisafinitegyrogroupwithtrivialgyroautomorphisms,thatis,Gisafinitegroup,
thenGactsonitselfbygroup-theoreticconjugation
a·x=(a⊕x)⊖a
foralla∈G,x∈G. Inthiscase,thesetoffixedpointsofGequalsZ(G), thegroup-
theoretic center of G. Further, the stabilizer of x equals C (x), the group-theoretic
G
centralizer of x in G. From the orbit decomposition theorem, we recover the class
equationinfinitegrouptheory,
n
(cid:229)
|G|=|Z(G)|+ [G:C (x)], (14)
G i
i=1
wherex ,x ,...,x arerepresentativesforthedistinctconjugacyclassesofGnotcon-
1 2 n
tainedinthecenterofG.
As a consequence of the orbit-stabilizer theorem, we prove the Burnside lemma,
alsoknownastheCauchy-Frobeniuslemma,forfinitegyrogroups.
LetX beaG-set. Recallthatforx∈X,thestabilizerofxinGisthesubgyrogroup
stabx={a∈G: a·x=x}.
Dually,fora∈G,thesetofelementsofX fixedbyaisdefinedas
fixa={x∈X: a·x=x}. (15)
Theorem 3.11 (The Burnside lemma). Let G be a finite gyrogroup and let X be a
finiteG-set. ThenumberofdistinctorbitsinX equals
1 (cid:229)
|fixa|.
|G|
a∈G
Proof. DefineY ={(a,x)∈G×X: a·x=x}. WecountthenumberofelementsofY
intwoways.
Notethatforeacha∈G, (a,x)∈Y ifandonlyifa·x=xifandonlyifx∈fixa.
Hence,thereareexactly|fixa|pairsinY withfirstcoordinatea. Itfollowsthat
(cid:229)
|Y|= |fixa|. (16)
a∈G
T. Suksumran Gyrogroup actions 9
(cid:229)
Dually, |Y|= |stabx|. Assume that X is partitionedinto n distinct orbits, namely
x∈X
orbx ,orbx ,...,orbx . Foreachx∈X,xbelongstoexactlyoneorbit,so
1 2 n
n
(cid:229) (cid:229)
|Y|= |stabx| . (17)
i=1 x∈orbxi !
Notethatxbelongstoorbx ifandonlyiforbx=orbx. By(11),|stabx|=|stabx|for
i i i
allx∈orbx. Hence,
i
(cid:229) (cid:229)
|stabx|= |stabx|=|orbx||stabx|=|G|.
i i i
x∈orbxi x∈orbxi
By(17),
n
(cid:229)
|Y|= |G|=n|G|. (18)
i=1
Equating(16)and(18)completestheproof.
Thefollowingresultsleadtoadeepunderstandingoftransitivegyrogroupactions,
whichwillbestudiedindetailinSections3.4and4.
Theorem3.12. LetX beaG-set. Foralla∈G,x∈X,
stab(a·x)={(a⊕c)⊟a: c∈stabx}.
Proof. For a, b∈G, x∈X, directcomputationshowsthatb∈stab(a·x)if andonly
if⊖a⊕(b⊕a)∈stabx. ThetheoremisanapplicationoftherightcancellationlawsI
andII.
Because oftheabsenceofassociativityin gyrogroups,the expressiona⊕b⊖ais
ambiguous.Hence,theconjugateofbbyacannotbedefinedasa⊕b⊖a,asingroup
theory. We formulate an appropriate notion of conjugate elements in a gyrogroup,
whichismotivatedbyTheorem3.12.
Definition3.13(Conjugates). Theelement(a⊕b)⊟aiscalledtheconjugateofbby
a. ForagivensubsetBofG,theconjugateofBbya,denotedbyconj (B),isdefined
a
by
conj (B)={(a⊕b)⊟a: b∈B}. (19)
a
Anelementaisconjugatetoanelementbifaistheconjugateofbbysomeelement
ofG. AsubsetAisconjugatetoasubsetBifAistheconjugateofBbysomeelement
ofG.
If G is a gyrogroup with trivial gyroautomorphisms, then a⊞b= a⊕b for all
a, b∈G and the gyrogroup operation ⊕ is associative. In this case, the notion of
conjugationinDefinition3.13reducestothatofgroup-theoreticconjugation.
Theorem3.14. LetX beaG-set. Forallx,y∈X,ifx∼y,thenstabxandstabyare
conjugatetoeachother.
T. Suksumran Gyrogroup actions 10
Proof. Supposethatx∼y. Theny=a·xforsomea∈G. ByTheorem3.12,
staby=stab(a·x)=conj (stabx).
a
Similarly,x=(⊖a)·yimpliesstabx=conj (staby).
⊖a
3.3 Invariantsubsets
LetGbeagyrogroupactingonX. ForagivensubsetY ofX,define
GY ={a·y: a∈G,y∈Y}. (20)
WesaythatY isaninvariantsubsetofX ifGY ⊆Y. Clearly,Y isaninvariantsubsetof
X ifandonlyifGY =Y. Foreachx∈X,orbxisindeedaninvariantsubsetofX.More
generally,ifY isanonemptysubsetofX,thenGY isaninvariantsubsetofX. Invariant
subsetsplaytheroleofsubstructuresofaG-set,asthefollowingpropositionindicates.
Proposition3.15. LetX beaG-set. IfY isaninvariantsubsetofX,thenGactsonY
bytherestrictionoftheactionofGtoY.
Proof. BecauseGY =Y,a·ybelongstoY foralla∈Gandy∈Y. Thattheaxiomsof
agyrogroupactionaresatisfiedisimmediate.
3.4 Types ofactions
Followingterminologyinthetheoryofgroupactions,wepresentseveraltypesof
gyrogroup actions. If G is a gyrogroup with gyroautomorphisms being the identity
automorphism,theterminologycorrespondstoitsgroupcounterpart.
Definition 3.16 (Faithful actions). A gyrogroup action of G on X is faithful if the
gyrogrouphomomorphismaffordedbytheactionisinjective.
Theorem3.17. LetX beaG-setandletp : G→Sym(X)betheassociatedpermuta-
tionrepresentation.ThequotientgyrogroupG/kerp actsfaithfullyonX by
(a⊕kerp )·x=a·x (21)
foralla∈G,x∈X.
Proof. Set K =kerp . By Lemma 3.8, (21) is well defined. Since G/K admits the
quotientgyrogroupstructure,wehave
(a⊕K)·((b⊕K)·x)=((a⊕K)⊕(b⊕K))·x
foralla,b∈G,x∈X. Thisprovesthat(21)definesanactionofG/KonX.
Letj˙ bethegyrogrouphomomorphismaffordedbytheaction(21). Supposethat
a⊕K∈G/K is such that j˙(a⊕K)=id . Hence, (a⊕K)·x=x for all x∈X. By
X
(21),a·x=xforallx∈X. Hence,a∈K andsoa⊕K=0⊕K. Thisprovesthatj˙ is
injectiveandhencetheaction(21)isfaithful.