Connected quandles associated with pointed abelian groups

33 Pages·2013·0.46 MB·English

Pacific
Journal of
Mathematics
CONNECTED QUANDLES ASSOCIATED
WITH POINTED ABELIAN GROUPS
W. EDWIN CLARK, MOHAMED ELHAMDADI, XIANG-DONG HOU,
MASAHICO SAITO AND TIMOTHY YEATMAN
Volume264 No.1 July2013
PACIFICJOURNALOFMATHEMATICS
Vol.264,No.1,2013
dx.doi.org/10.2140/pjm.2013.264.31
CONNECTED QUANDLES ASSOCIATED
WITH POINTED ABELIAN GROUPS
W. EDWIN CLARK, MOHAMED ELHAMDADI, XIANG-DONG HOU,
MASAHICO SAITO AND TIMOTHY YEATMAN
A quandle is a self-distributive algebraic structure that appears in quasi-
groupandknottheories. Foreachabeliangroup Aand c∈ A,wedefinea
quandleG(A,c)on(cid:90) ×A. Thesequandlesaregeneralizationsofaclassof
3
nonmedialLatinquandlesdefinedbyV.M.Galkin,sowecallthemGalkin
quandles. Each G(A,c) is connected but not Latin unless A has odd or-
der. G(A,c) is nonmedial unless 3A = 0. We classify their isomorphism
classes in terms of pointed abelian groups and study their various proper-
ties. AfamilyofsymmetricconnectedquandlesisconstructedfromGalkin
quandles, and some aspects of knot colorings by Galkin quandles are also
discussed.
1. Introduction
Sets with certain self-distributive operations called quandles have been studied
since the 1940s in various areas. They have been studied, for example, as an
algebraicsystemforsymmetries[Takasaki1943],asquasigroups[Galkin1988],
andinrelationtomodules[Nelson2003]. Thefundamentalquandlewasdefined
inamannersimilarto thefundamentalgroup[Joyce1982;Matveev1982], which
madequandlesanimportanttoolinknottheory. Algebraichomologytheoriesfor
quandlesweredefined [Carteretal.2003b; Fennetal. 1995]anddevelopedand
investigated([LitherlandandNelson2003;Mochizuki2011;Niebrzydowskiand
Przytycki2009;2011;Nosaka2011],forexample),andextensionsofquandlesby
cocycleshavebeenstudied[AndruskiewitschandGraña2003;Carteretal.2003a;
Eisermann2007b]andappliedtovariouspropertiesofknotsandknottedsurfaces
(see[Carteretal.2004]andreferencestherein).
Beforealgebraictheoriesofextensionsweredeveloped, Galkin[1988]defineda
familyofquandlesthatareextensionsofthe3-elementconnectedquandle R ,and
3
wecallthemGalkinquandles. EventhoughthedefinitionofGalkinquandlesisa
M.S.wassupportedinpartbyNSFgrantDMS0900671.
MSC2010: 57M25.
Keywords: quandles,pointedabeliangroups,knotcolorings.
31
32 CLARK,ELHAMDADI,HOU,SAITOANDYEATMAN
specialcaseofacocycleextensiondescribedin[AndruskiewitschandGraña2003],
theyhavecuriouspropertiessuchastheexplicitandsimpledefiningformula,close
connectionsto dihedralquandles,and thefactthattheyappearinthe listofsmall
connectedquandles.
Inthispaper,wegeneralizeGalkin’sdefinitionanddefineafamilyofquandles
thatareextensionsof R ,characterizetheirisomorphismclasses,andstudytheir
3
properties. The definition is given in Section 3 after a brief review of necessary
materialsinSection2. Isomorphismclassesarecharacterizedbypointedabelian
groupsinSection4. VariousalgebraicpropertiesofGalkinquandlesareinvestigated
inSection5,andtheirknotcoloringsarestudiedinSection6.
2. Preliminaries
In thissection we briefly review somedefinitions and examplesof quandles. More
details can be found, for example, in [Andruskiewitsch and Graña 2003; Carter
etal.2004;Fennetal.1995].
A quandle X is a set with a binary operation (a,b) (cid:55)→ a ∗b satisfying the
followingconditions.
(1) (Idempotency) Foranya∈ X,a∗a=a.
(2) (Invertibility) Foranyb,c∈ X,
thereisauniquea∈ X suchthata∗b=c.
(3) (Rightself-distributivity) Foranya,b,c∈ X,
wehave(a∗b)∗c=(a∗c)∗(b∗c).
Aquandlehomomorphismbetweentwoquandles X,Y isamap f : X →Y such
that f(x∗ y)= f(x)∗ f(y),where∗ and∗ denotethequandleoperationsof X
X Y X Y
andY,respectively. Aquandleisomorphismisabijectivequandlehomomorphism,
andtwoquandlesareisomorphicifthereisaquandleisomorphismbetweenthem.
Typicalexamplesofquandlesincludethefollowing.
• Anynonemptyset X withtheoperationx∗y=x foranyx,y∈X isaquandle
calledthetrivialquandle.
• Agroup X =G withtheoperationofn-foldconjugation,a∗b=b−nabn,isa
quandle.
• Letn beapositiveinteger. Fora,b∈(cid:90) (integersmodulon),define
n
a∗b≡2b−a (modn).
Then ∗ defines a quandle structure called the dihedralquandle R . This set
n
canbeidentifiedwiththesetofreflectionsofaregularn-gonwithconjugation
asthequandleoperation.
CONNECTEDQUANDLESASSOCIATEDWITHPOINTEDABELIANGROUPS 33
• Any(cid:90)[T,T−1]-module M isaquandlewitha∗b=Ta+(1−T)bfora,b∈M.
ThisiscalledanAlexanderquandle. AnAlexanderquandleisalsoregarded
asapair(M,T),where M isanabeliangroupand T ∈Aut(M).
Let X beaquandle. TherighttranslationR : X → X bya∈ X isdefinedby
a
R (x)=x∗aforx∈X. Similarly,thelefttranslationL isdefinedbyL (x)=a∗x.
a a a
ThenR isapermutationof X byAxiom(2). ThesubgroupofSym(X)generated
a
bythepermutationsR ,a∈ X,iscalledtheinnerautomorphismgroupof X and
a
isdenotedbyInn(X). Welistsomedefinitionsofcommonlyknownpropertiesof
quandlesbelow.
• AquandleisconnectedifInn(X)actstransitivelyon X.
• ALatinquandleisaquandlesuchthatforeacha∈ X,thelefttranslationL
a
isabijection. Thatis,themultiplicationtableofthequandleisaLatinsquare.
• Aquandleisfaithfulifthemappinga(cid:55)→R isaninjectionfrom X toInn(X).
a
• A quandle X is involutory, or a kei, if the right translations are involutions:
R2 =idforalla∈ X.
a
• Theoperation∗¯ on X definedbya ∗¯ b=R−1(a)isaquandleoperation,and
b
(X,∗¯)iscalledthedualquandleof(X,∗). If(X,∗¯)isisomorphicto(X,∗),
then(X,∗)iscalledself-dual.
• Aquandle X ismedialif(a∗b)∗(c∗d)=(a∗c)∗(b∗d)foralla,b,c,d∈X.
It is also called abelian. It is known and easily seen that every Alexander
quandleismedial.
Acoloringofanorientedknotdiagrambyaquandle X isamapC:A→X from
thesetofarcsAofthediagramto X suchthattheimageofthemapsatisfiesthe
relationdepictedinFigure1ateachcrossing. Moredetailscanbefoundin[Carter
et al. 2004; Eisermann 2007a], for example. A coloring that assigns the same
elementof X forallthearcsiscalledtrivial,andotherwisenontrivial. Thenumber
ofcoloringsofaknotdiagrambyafinitequandleisknowntobeindependentofthe
choiceofadiagram,andhenceisaknotinvariant. Acoloringbyadihedralquandle
R for a positive integer n > 1 is called an n-coloring. If a knot is nontrivially
n
colored by a dihedral quandle R for a positive integer n > 1, then it is called
n
n-colorable. InFigure2,anontrivial3-coloringofthetrefoilknot(3 inacommon
1
notationinaknottable[ChaandLivingston2011])isindicated. Thisispresented
y
x x∗y
Figure1. Acoloringruleatacrossing.
34 CLARK,ELHAMDADI,HOU,SAITOANDYEATMAN
0 1
2
0
1
Figure2. Trefoilastheclosureofσ3.
1
inaclosedbraidform. Eachcrossingcorrespondstoastandardgeneratorσ ofthe
1
2-strand braid group, and σ3 represents three crossings together as in the figure.
1
Thedottedlineindicatestheclosure;see[Rolfsen1976]formoredetailsofbraids.
Thefundamentalquandleisdefinedinamannersimilartothefundamentalgroup
[Joyce1982;Matveev1982]. Apresentationofaquandleisdefinedinamanner
similartogroupsaswell,andapresentationofthefundamentalquandleisobtained
from a knot diagram (see, for example, [Fenn and Rourke 1992]), by assigning
generators to arcs of a knot diagram, and relations corresponding to crossings.
The set of colorings of a knot diagram K by a quandle X is then in one-to-one
correspondence with the set of quandle homomorphisms from the fundamental
quandleof K to X.
3. DefinitionandnotationforGalkinquandles
Let A beanabeliangroup,alsoregardednaturallyasa(cid:90)-module. Letµ:(cid:90) →(cid:90),
3
τ :(cid:90) → A be functions. These functions µ and τ need not be homomorphisms.
3
Defineabinaryoperationon(cid:90) ×A by
3
(x,a)∗(y,b)=(cid:0)2y−x,−a+µ(x−y)b+τ(x−y)(cid:1), x,y∈(cid:90) , a,b∈ A.
3
Proposition 3.1. For any abelian group A, the operation ∗ defines a quandle
structureon(cid:90) ×Aifµ(0)=2,µ(1)=µ(2)=−1,andτ(0)=0.
3
Galkin[1988,p.950]gavethisdefinitionfor A=(cid:90) . Thepropositiongeneralizes
p
hisresulttoanyabeliangroup A. Fortheproof,weexaminetheaxioms.
Lemma3.2. (A) Theoperationisidempotent—thatis,itsatisfiesAxiom(1)—if
andonlyif(µ(0)−2)a=0foranya∈ A,andτ(0)=0.
(B) Theoperationasarightactionisinvertible—thatis,itsatisfiesAxiom(2).
Proof. Directcalculations. (cid:3)
CONNECTEDQUANDLESASSOCIATEDWITHPOINTEDABELIANGROUPS 35
Lemma3.3. Theoperation∗on(cid:90) ×Aisrightself-distributive—thatis,itsatisfies
3
Axiom(3)—ifandonlyifµ,τ satisfythefollowingconditionsforany X,Y ∈(cid:90)
3
andb,c∈ A:
(4) µ(−X)b=µ(X)b,
(5) (cid:0)µ(X+Y)+µ(X−Y)(cid:1)c=(µ(X)µ(Y))c,
(6) τ(X+Y)+τ(Y −X)=τ(X)+τ(−X)+µ(X)τ(Y).
Proof. Rightself-distributivity,thatis,
((x,a)∗(y,b))∗(z,c)=((x,a)∗(z,c))∗((y,b)∗(z,c))
for x,y,z∈(cid:90) anda,b,c∈ A,issatisfiedifandonlyif
3
µ(x−y)b=µ(y−x)b,
µ(2y−x−z)c=(cid:0)−µ(x−z)+µ(y−x)µ(y−z)(cid:1)c,
−τ(x−y)+τ(2y−x−z)=−τ(x−z)+µ(y−x)τ(y−z)+τ(y−x).
Thisisseenbyequatingthecoefficientsofb andc andtheconstantterm. Forthe
equivalence of the first equation with (4), set X =x−y. For the equivalence of
thesecondwith(5),set X = y−x and Y =z−y. Fortheequivalenceofthelast
with(6),set X = y−x andY = y−z. (cid:3)
ProofofProposition3.1. Assumetheconditionsstated. ByLemma3.2,Axioms(1)
and (2) are satisfied under the specifications µ(0) = 2,µ(1) = µ(2) = −1, and
τ(0)=0.
If X =0orY =0,then(5)(togetherwith(4))becomesatautology. If X−Y =0
or X +Y =0, then (5) reduces to µ(2X)+2=µ(X)2, which is satisfied by the
above specifications. For R , if X +Y (cid:54)=0 and X −Y (cid:54)=0, then either X =0 or
3
Y =0. Hence (5) is satisfied. For (6), it is checked similarly, for the two cases
[X =0orY =0]and[X−Y =0or X+Y =0]. (cid:3)
Definition3.4. Let A beanabeliangroup. Thequandledefinedby∗on(cid:90) ×A by
3
Proposition3.1,
(x,a)∗(y,b)=(cid:0)2y−x,−a+µ(x−y)b+τ(x−y)(cid:1), x,y∈(cid:90) , a,b∈ A,
3
withµ(0)=2,µ(1)=µ(2)=−1,andτ(0)=0,iscalledtheGalkinquandleand
denotedby G(A,τ).
Sinceτ isspecifiedbythevaluesτ(1)=c andτ(2)=c wherec ,c ∈ A,we
1 2 1 2
alsodenoteitby G(A,c ,c ).
1 2
Example3.5. TheGalkinquandle G((cid:90) ,0,1)is(cid:90) ×(cid:90) asasetwiththequandle
2 3 2
operationdefinedasabovewithµ(0)=2,µ(1)=µ(2)=−1,τ(0)=τ(1)=0,and
36 CLARK,ELHAMDADI,HOU,SAITOANDYEATMAN
τ(2)=1. Thus,(0,1)∗(1,0)=(2,−1+µ(2)0+τ(2))=(2,0)and(2,0)∗(1,1)=
(0,0+µ(1)1+τ(1))=(0,1),forexample.
Lemma 3.6. For any abelian group A and c ,c ∈ A, the quandles G(A,c ,c )
1 2 1 2
andG(A,0,c −c )areisomorphic.
2 1
Proof. Letc=c −c . Defineη:G(A,c ,c )→G(A,0,c),asamapon(cid:90) ×A,by
2 1 1 2 3
η(x,a)=(x,a+β(x))whereβ(0)=β(1)=0andβ(2)=−c . Thisηisabijection,
1
and we show that it is a quandle homomorphism. We compute η((x,a)∗(y,b))
andη(x,a)∗η(y,b)for x,y∈(cid:90) anda,b∈ A.
3
Ifx=y,thenµ(x−y)=2andτ(x−y)=0forbothG(A,c ,c )andG(A,0,c),
1 2
sothat
η((x,a)∗(x,b))=η(x,2b−a)=(x,2b−a+β(x)),
η(x,a)∗η(x,b)=(x,a+β(x))∗(x,b+β(x))=(cid:0)x,2(b+β(x))−(a+β(x))(cid:1)
=(x,2b−a+β(x)),
asdesired.
If x−y=1∈(cid:90) ,thenµ(x−y)=−1forbothG(A,c ,c )andG(A,0,c)and
3 1 2
τ(x−y)=c for G(A,c ,c )butτ(x−y)=0for G(A,0,c),sothat
1 1 2
η((x,a)∗(y,b))=η(2y−x,−a−b+c )=(2y−x,−a−b+c +β(2y−x)),
1 1
η(x,a)∗η(y,b)=(x,a+β(x))∗(y,b+β(y))
=(cid:0)2y−x, −(a+β(x))−(b+β(y))(cid:1).
Thetwoexpressionsareequalifandonlyifβ(x)+β(y)+β(2y−x)=−c ,which
1
istruesince x (cid:54)= y impliesthatexactlyoneof x,y,2y−x is2∈(cid:90) .
3
If x−y=2∈(cid:90) ,thenµ(x−y)=−1forbothG(A,c ,c )andG(A,0,c)and
3 1 2
τ(x−y)=c for G(A,c ,c )butτ(x−y)=c −c =c for G(A,0,c),sothat
2 1 2 2 1
η((x,a)∗(y,b))=η(2y−x,−a−b+c )=(2y−x,−a−b+c +β(2y−x)),
2 2
η(x,a)∗η(y,b)=(x,a+β(x))∗(y,b+β(y))
=(cid:0)2y−x,−(a+β(x))−(b+β(y))+c −c (cid:1)
2 1
=(cid:0)2y−x,−a−b−β(x)−β(y)+(c −c )(cid:1),
2 1
andagaintheseareequalforthesamereasonasabove. (cid:3)
Notation. Since,byLemma3.6,anyGalkinquandleisisomorphicto G(A,0,c)
foranabeliangroup A andc∈ A,wedenote G(A,0,c)by G(A,c)forshort.
Anyfiniteabeliangroupisaproduct(cid:90) ×···×(cid:90) ,wherethepositiveintegers
n1 nk
nj satisfynj|nj+1 for j=1,...,k−1. Inthiscase,anyelementc∈Aiswrittenin
avectorform[c ,...,c ],wherec ∈(cid:90) . ThenthecorrespondingGalkinquandle
1 k j nj
isdenotedby G(A,[c ,...,c ]).
1 k
CONNECTEDQUANDLESASSOCIATEDWITHPOINTEDABELIANGROUPS 37
Remark3.7. WenotethatthedefinitionofGalkinquandlesinducesafunctor. Let
Ab denote the category of pointed abelian groups; its objects are pairs (A,c),
0
where A is an abelian group and c ∈ A, and its morphisms f :(A,c)→(B,d)
aregrouphomomorphisms f : A→ B suchthat f(c)=d. LetQbethecategory
of quandles consisting of quandles as objects and quandle homomorphisms as
morphisms.
F
Thenthecorrespondence(A,c)(cid:55)→G(A,c)definesafunctorF:Ab →Q. It
0
iseasytoverifythatifamorphism f :(A,c)→(B,d)isgiven,thenthemapping
F(f)(x,a)=(x, f(a))with(x,a)∈G(A,c)=(cid:90) ×A isahomomorphismfrom
3
G(A,c)to G(B,d)andsatisfiesF(gf)=F(g)F(f)andF(id(A,c))=idG(A,c).
Remark3.8. AreaderwillwondertowhatextentDefinition3.4ofaGalkinquandle
canbegeneralized. Wetriedseveralgeneralizations. Forexample,ifoneattempts
toreplace3byanarbitraryprime p inDefinition3.4,thenLemma3.3stillholds.
Inthiscasefor p>3,weproveinLemma5.14thatµ(x)=2forall x ∈(cid:90) ,and
p
computer experiments indicate that one almost always obtains a quandle if and
onlyifτ =0,inwhichcasethequandleobtainedissimplyaproductofdihedral
quandles. We have also attempted to replace −x+2y by the Alexander quandle
operationtx+(1−t)y inboththeleftandrightcoordinates,buthaveneitherbeen
successfulinfindinginterestingnewquandles,norbeenabletoprovethatnosuch
generalizations exist. We note that if a generalization for p >3 exists, then any
such quandles will be less dense than Galkin quandles, since multiples of 3 are
morenumerousthanmultiplesof p when p>3.
4. Isomorphismclasses
InthissectionweclassifyisomorphismclassesofGalkinquandles.
Lemma4.1. Let Abeanabeliangroup,andleth:A→A(cid:48)beagroupisomorphism.
ThenGalkinquandlesG(A,τ)andG(A(cid:48),hτ)areisomorphicasquandles.
Proof. Define f : G(A,τ) → G(A(cid:48),hτ), as a map from (cid:90) × A to (cid:90) × A(cid:48),
3 3
by f(x,a) = (x,h(a)). This f is a bijection, and we show that it is a quandle
homomorphismbycomputing f((x,a)∗(y,b))and f(x,a)∗ f(y,b)forx,y∈(cid:90)
3
anda,b∈ A:
f((x,a)∗(y,b))= f(2y−x,−a+µ(x−y)b+τ(x−y))
=(cid:0)2y−x,h(−a+µ(x−y)b+τ(x−y))(cid:1),
f(x,a)∗ f(y,b)=(x,h(a))∗(y,h(b))
=(cid:0)2y−x,−h(a)+µ(x−y)h(b)+hτ(x−y)(cid:1).
Theequality f((x,a)∗(y,b))= f(x,a)∗ f(y,b)followsfromthefactsthat h is
agrouphomomorphismandµ(x−y)isaninteger. (cid:3)
38 CLARK,ELHAMDADI,HOU,SAITOANDYEATMAN
Lemma 4.2. Let c,d,n be positive integers. If gcd(c,n) = d, then G((cid:90) ,c) is
n
isomorphictoG((cid:90) ,d).
n
Proof. If A = (cid:90) , then Aut(A) = (cid:90)∗ = unitsof(cid:90) , and the divisors of n are
n n n
representativesoftheorbitsof(cid:90)∗ actingon(cid:90) . (cid:3)
n n
Thus we may choose the divisors of n for the values of c for representing
isomorphismclassesof G((cid:90) ,c).
n
Corollary4.3. If Aisavectorspace(elementary p-group),thenthereareexactly
twoisomorphismclassesofGalkinquandlesG(A,τ).
Proof. If A isavectorspacecontainingnonzerovectorsc andc ,thenthereisa
1 2
nonsingularlineartransformation h of A suchthat h(c )=c . That G(A,0)isnot
1 2
isomorphicto G(A,c)ifc(cid:54)=0followsfromLemma4.5. (cid:3)
For distinguishing isomorphismclasses, cycle structures of the right action are
useful,andweusethefollowinglemmas.
Lemma4.4. Foranyabeliangroup A,theGalkinquandleG(A,τ)isconnected.
Proof. Recallthattheoperationisdefinedbytheformula
(x,a)∗(y,b)=(cid:0)2y−x,−a+µ(x−y)b+τ(x−y)(cid:1),
with µ(0)=2, µ(1)=µ(2)=−1, and τ(0)=0. If x (cid:54)= y, then (x,a)∗(y,b)=
(2y−x,−a−b+c )=(z,c),wherei =1or2and x,y∈(cid:90) anda,b∈ A. Note
i 3
that{x,y,2y−x}=(cid:90) if x (cid:54)=y. Inparticular,forany(x,a)and(z,c)with x (cid:54)=z,
3
thereis(y,b)suchthat(x,a)∗(y,b)=(z,c).
Forany(x,a )and(x,a )wherex∈(cid:90) anda ,a ∈A,take(z,c)∈(cid:90) ×Asuch
1 2 3 1 2 3
thatz(cid:54)=x. Thenthereare(y,b ),(y,b )suchthatx(cid:54)=y(cid:54)=zand(x,a )∗(y,b )=
1 2 1 1
(z,c)and(z,c)∗(y,b )=(x,a ). Hence G(A,τ)isconnected. (cid:3)
2 2
Lemma4.5. Thecyclestructureofarighttranslationin G(A,τ),whereτ(0)=
τ(1)=0andτ(2)=c,consistsof1-cycles,2-cycles,and2k-cycles,wherek isthe
orderofcinthegroup A.
Sinceisomorphicquandleshavethesamecyclestructureofrighttranslations,
G(A,c)andG(A,c(cid:48))forc,c(cid:48)∈ Aarenotisomorphicunlesstheordersofcandc(cid:48)
coincide.
Proof. Let τ(0) = 0, τ(1) = 0, and τ(2) = c. Then by Lemma 4.4, the cycle
structureofeachcolumnisthesameasthe cyclestructureoftherighttranslation
by(0,0),thatis,ofthepermutation f(x,a)=(x,a)∗(0,0)=(−x,−a+τ(x)).
Weshowthatthispermutationhascyclesoflengthonly1,2andtwicetheorder
ofc in A. Since f(0,a)=(0,−a)fora∈ A,a(cid:54)=0,wehave f2(0,a)=(0,a),so
that(0,a)generatesa2-cycle,ora1-cycleif2a=0. Nowfrom f(1,a)=(2,−a)
and f(2,a)=(1,−a+c) fora∈ A, by induction it is easy to see that for k >0,
CONNECTEDQUANDLESASSOCIATEDWITHPOINTEDABELIANGROUPS 39
f2k(1,a)=(1,a+kc)and f2k(2,a)=(2,a−kc). Inthecaseof(1,a),a(cid:54)=0,the
cyclecloseswhena+kc=a in A. Thesmallestk forwhichthisholdsistheorder
ofc,inwhichcasethecycleisoflength2k. Acyclebeginningat(2,a)similarly
hasthissamelength. (cid:3)
Proposition4.6. Letn beapositiveinteger. Let A=(cid:90) andc ,c(cid:48) ∈(cid:90) fori =1,2.
n i i n
TwoGalkinquandles G(A,c ,c )and G(A,c(cid:48),c(cid:48))areisomorphicifandonlyif
1 2 1 2
gcd(c −c ,n)=gcd(c(cid:48) −c(cid:48),n).
1 2 1 2
Proof. Ifgcd(c −c ,n)=gcd(c(cid:48) −c(cid:48),n),thentheyareisomorphicbyLemmas3.6
1 2 1 2
and4.2. Thecyclestructuresaredifferentifgcd(c −c ,n)(cid:54)=gcd(c(cid:48) −c(cid:48),n)by
1 2 1 2
Lemma4.5,andhencetheyarenotisomorphic. (cid:3)
Remark 4.7. The cycle structure is not sufficient for noncyclic groups A. For
example,let A=(cid:90) ×(cid:90) . Then G(A,[1,0])and G(A,[0,2])havethesamecycle
2 4
structure for right translations, with cycle lengths {2,2,4,4,4,4} in a multiset
notation,yettheyareknownnottobeisomorphic. (InthenotationofExample4.12
below, G(A,[1,0])=C[24,29]and G(A,[0,2])=C[24,31].) Wenotethatthere
isnoautomorphismof A carrying[1,0]to[0,2].
Moregenerally,theisomorphismclassesofGalkinquandlesarecharacterizedas
follows.
Theorem 4.8. Suppose A,A(cid:48) are finite abelian groups. Two Galkin quandles
G(A,τ) and G(A(cid:48),τ(cid:48)) are isomorphic if and only if there exists a group isomor-
phismh: A→ A(cid:48) suchthathτ =τ(cid:48).
OneimplicationintheproofofTheorem4.8isLemma4.1. Fortheother,first
weprovethefollowingtwolemmas. Wewilluseawellknowndescriptionofthe
automorphismsofafiniteabeliangroup,whichcanbefoundin[HillarandRhea
2007;Ranum1907].
Lemma 4.9. Let A be a finite abelian p-group and let f : pA → pA be an
automorphism. Then f canbeextendedtoanautomorphismof A.
Proof. Let A=(cid:90)n1 ×···×(cid:90)nk. Then
p1 pk
     
px px x
2 2 2
. . .
(7) f  .. = P .. ,  .. ∈(cid:90)n2 ×···×(cid:90)nk,
      p2 pk
px px x
k k k
where
 P P ··· P 
22 23 2k
 pP32 P33 ··· P3k
(8) P = . . . ,
 .. .. .. 
 
pk−2P pk−3P ··· P
k2 k3 kk