Commutativity in non-Abelian Groups

34 Pages·2010·0.75 MB·English

Commutativity in non-Abelian Groups
Cody Clifton
May 6, 2010
Abstract.
Let P (G) be defined as the probability that any two elements selected at ran-
2
dom from the group G, commute with one another. If G is an Abelian group,
P (G) = 1, so our interest lies in the properties of the commutativity of non-
2
Abelian groups.
Particular results include that the maximum commutativity of a non-Abelian
group is 5/8, and this degree of commutativity only occurs when the order of
thecenterofthegroupisequaltoonefourththeorderofthegroup. Explicitex-
ampleswillbeprovidedofarbitrarilylargenon-Abeliangroupsthatexhibitthis
maximumcommutativity,alongwithaproofthatthereareno5/8commutative
groups of order 4mod8.
Further,weprovethatnogroupexhibitscommutativity0,thereexistexamples
ofgroupswhosecommutativityisarbitrarilycloseto0. Then, weshowthatfor
every positive integer n there exists a group G such that P (G)=1/n. Finally
2
we prove that the commutativity of a factor group G/N of a group G is always
greater than or equal to the commutativity of G.
0 Introduction:
ThewaywedefineAbeliangroups(seeDefinitions0.2and0.3)providesuswith
asimplewayofunderstandingwhatmightbetermedthecommutativityofsuch
groups. In particular, as each pair of elements in an Abelian group necessarily
commutes, we can say that the group has complete commutativity, or 100%
commutativity, or, on a decimal scale of zero to one, commutativity 1. As is
known by those with even a little background in the study of group theory,
however, Abelian groups account for only a small proportion of all groups.
1
Given that these other, so-called non-Abelian groups, represent the vast major-
ity of all groups, we would like to develop a method for classifying the degree,
or the percent, to which a given non-Abelian group is commutative. As we
will see directly from the following definitions, even non-Abelian groups cannot
havecommutativity0onthezerotoonescale,sinceatleastonegroupelement,
namely the identity (see Definition 0.2), will always commute with every other
element. Also, as a non-Abelian group is, by definition, not Abelian, it must
contain some elements that do not commute, so we cannot have a non-Abelian
group with commutativity 1.
Since, at a glance, we already can rule out 0 and 1 as possible values for the
commutativity of a given non-Abelian group, our initial investigations will con-
sist primarily of showing what values, between 0 and 1, are possible degrees
of non-Abelian commutativity. In addition, we will spend much of our time
describing the ways in which many the characteristics of commutativity are ex-
hibited by finite non-Abelian groups, such as the dihedral groups. Finally, we
will conclude by demonstrating additional properties of the commutativity of
non-Abelian groups, such as the relationship between the commutativity of a
group, and the commutativity of one of its factor groups.
To begin our investigation of the commutativity of non-Abelian groups we will
first provide a few definitions. Unless otherwise specified, all notation and ter-
minology used will be as in [1].
Definition 0.1: Binary Operation.
LetGbeaset. Abinary operation onGisafunctionthatassignseachordered
pair of elements of G an element of G.
Fromthedefinitionofabinaryoperation,wenextdevelopthenotionofagroup.
Definition 0.2: Group
LetGbeanonemptysettogetherwithabinaryoperation(usuallycalledmulti-
plication)thatassignstoeachorderedpair(a,b)ofelementsofGanelementof
Gdenotedbyab. WesaythatGisagroup underthisoperationifthefollowing
three properties are satisfied.
1. Associativity. The operation is associative; that is, (ab)c=a(bc)
for all a, b, c in G.
2. Identity. There is an element e (called the identity) in G such
that ae=ea=a for all a in G.
3. Inverses. For each element a in G, there is an element b in G
(called the inverse of a) such that ab=ba=e.
Now, to characterize the difference between Abelian and non-Abelian groups,
we provide the following definition.
2
Definition 0.3: Abelian Group
If a group has the property that ab=ba for every pair of elements a and b, we
say that the group is Abelian. A group is non-Abelian if there is some pair of
elements a and b for which ab(cid:54)=ba.
Finally, the next two definitions require particular attention, as they will be
directly referred to numerous times throughout this article.
Definition 0.4: Center of a Group
The center, Z(G), of a group G is the subset of elements in G that commute
with every element of G. In symbols, Z(G)={a∈G | ax=xa for all x in G}.
Definition 0.5: Centralizer of a in G
LetabeafixedelementofagroupG. Thecentralizer of a in G,C(a),istheset
ofallelementsinGthatcommutewitha. Insymbols,C(a)={g ∈G|ga=ag}.
In addition to the five definitions above, we will later refer to both the Class
Equation and Lagrange’s Theorem, both of which will be stated and explained
prior to their usage.
1 Commutativity
Using the definition of the center of a group and of the centralizer of a group
element, we are prepared to begin developing a measure for the commutativity
of a given group. First, though, we define a certain family of groups.
Dihedral groups
The following definition describes the family of groups that will play the most
important role in providing examples of the properties of non-Abelian commu-
tativity throughout the remainder of this article.
Definition 1.1: Dihedral group 1
ThedihedralgroupD isthesymmetrygroupofann-sidedregularpolygonfor
n
n>1. The group order of D is 2n.
n
1See [4] for definition. A thorough explanation of the properties and construction of the
dihedralgroupscanbefoundin[1].
3
Byorder wemeanthenumberofelementsofagroup(finiteorinfinite). Wewill
use|G|todenotetheorderofagroupG. Now,tofurtherexplainthisdefinition,
we will consider as an example the dihedral group, D , of order eight.2
4
Fromthedefinition,weknowthatthedihedralgroupD isthesymmetrygroup
4
ofa4-sidedregularpolygon,thatis,asquare. Toelaborate,thismeansthatthe
group D consists of the eight flips and rotations which comprise the natural
4
transformations that can be performed on a square. The four rotations in the
plane are by degrees of 0, 90, 180, and 270. These are denoted R , R , R ,
0 90 180
and R respectively. The four remaining elements are two flips, H and V,
270
across the horizontal and vertical axes, and two flips, D and D(cid:48), across the two
diagonal axes.
These eight transformation elements are shown in Figure 1.
Figure 1: D ={R ,R ,R ,R ,H,V,D,D(cid:48)}
4 0 90 180 270
Nowthatwehavegainedaknowledgeoftheelementsthatcomprisethedihedral
groupD webegintoconsiderproductsoftheseelementsindifferentorders, so
4
as to determine how many pairs of elements commute out of the total number
of possible commutative pairs.
2ThereadershouldnotethatthepropertiesofthisparticulargroupD4 willbeinvaluable
in later sections of this article, as they exhibit many of the properties of commutativity of
non-Abeliangroupsthatwillbediscussedandproved.
4
The easiest way to summarize the result of taking the product of each pair
of elements in the group is to construct a Cayley table for D [see Table 1].
4
A Cayley table for group elements can be thought of in the same way as a
multiplication table for integers. That is, to determine the value of the product
ij, one may refer to the value in the intersection of the ith row and the jth
column.
D R R R R H V D D(cid:48)
4 0 90 180 270
R R R R R H V D D(cid:48)
0 0 90 180 270
R R R R R D(cid:48) D H V
90 90 180 270 0
R R R R R V H D(cid:48) D
180 180 270 0 90
R R R R R D D(cid:48) V H
270 270 0 90 180
H H D V D(cid:48) R R R R
0 180 90 270
V V D(cid:48) H D R R R R
180 0 270 90
D D V D(cid:48) H R R R R
270 90 0 180
D(cid:48) D(cid:48) H D V R R R R
90 270 180 0
Table 1: Cayley table for the group D
4
Asanexample, noticethatHR =D. Further, todemonstratethatthegroup
90
D isnon-Abelian,wemaynotethatR H =D(cid:48). Itwillbelefttothereaderto
4 90
verifythattheelementsD andD(cid:48) doindeedrepresenttheresultoftheproduct
of the elements H and R in the orders shown above.
90
A measure of commutativity
Now that we have one family of non-Abelian groups (the dihedral groups) to
examine, wecanbegintodevelopameasureforthecommutativityofgroupsin
general.
There are at least two ways to approach the problem of specifying a measure
of commutativity for non-Abelian groups. One uses the simple definition of the
centralizer of each element of the group, while the other uses the notions of
conjugates and conjugacy classes.
As it turns out, both of these methods lead to the same measure of commu-
tativity. We will discuss both methods, initially focusing on the perhaps more
intuitiveapproachinvolvingcentralizers,andlaterdetailingthesomewhatmore
elegant conjugacy class idea.
5
Henceforth, we will refer to the commutativity of a group G as P (G) (notice
2
that this notation applies even when the group G is Abelian, but that P (G)
2
trivially equals 1 if this is the case). The rationale behind this notation is that
the commutativity of a group as we will be measuring it is the probability that
any two group elements selected at random commute with one another, hence:
P .
2
Before defining the P measure of commutativity in terms of the centralizers of
2
thegroupelements,wenotethatthecenterofagroupbyitselfdoesnotprovide
aparticularlyaccuratemeasureforcommutativity. Inordertodemonstratethis
fact, we introduce some of the other small dihedral groups, specifically D , D ,
3 5
and D .3
6
As previously defined, the small dihedral groups D , D , and D consist of
3 5 6
elementscorrespondingtotherotation-andflip-symmetriesofregularpolygons
with 3, 5, and 6 sides respectively. In particular, R represents a rotation by
m
m degrees, and the other variables V,H,D,F and their primes represent flips
across the axes that run either from one vertex to the opposite vertex or from
the midpoint of one side to the midpoint of the opposite side (in the case of the
even-sidedpolygonssuchasthehexagon)orfromonevertextothemidpointof
the side opposite it (in the case of the odd-sided polygons such as the triangle
and the pentagon).
Thus, including D as defined earlier, we have:
4
D ={R ,R ,R ,V,D,D(cid:48)},
3 0 120 240
D ={R ,R ,R ,R ,V,H,D,D(cid:48)},
4 0 90 180 270
D ={R ,R ,R ,R ,R ,V,D,D(cid:48),F,F(cid:48)},
5 0 72 144 216 288
D ={R ,R ,R ,R ,R ,R ,V,V(cid:48),V(cid:48)(cid:48),D,D(cid:48),D(cid:48)(cid:48)},
6 0 60 120 180 240 300
where, in each case, R is the identity element of the group. Now, examining
0
the Cayley table of each group,4 we can conclude that the centers of the four
dihedral groups are:
Z(D )={R },
3 0
Z(D )={R ,R },
4 0 180
Z(D )={R },
5 0
Z(D )={R ,R }.
6 0 180
3Ingeneral,wewillnotconsiderthefirstdihedralgroupD2,asitonlyconsistsof4elements
andistherebyAbelian. Infact,whereas[4]definesthedihedralgroupsasbeginningwithD2,
[1]onlyconsidersthedihedralgroupsDn forn≥3.
4The reader may consult Table 1 for an example of the Cayley table of D4. The Cayley
tableofeachoftheotherthreegroupscanbeeasilyconstructed.
6
By the regular structural nature of the polygons that comprise the D groups,
n
we can conclude that when n is even the two elements in the center of D are
n
R and R , whereas when n is odd, the only element in Z(D ) is R . To see
0 180 n 0
this, we make note of the following facts.
Clearly R , a rotation by zero degrees, is the identity of every dihedral group
0
and is thus in the center of D for all n. Then, in all dihedral groups, every
n
rotation element commutes with every other rotation element, whereas every
flip element does not commute with most rotation elements. Since in the case
of even-sided polygon dihedral groups, every flip element commutes with the
identity, R , and the element R , but not with any other rotation element,
0 180
we conclude the Z(D )={R ,R } when n is even. On the other hand, every
n 0 180
flipelementinanodd-sidedpolygondihedralgroupdoesnotcommutewithany
rotation element besides the identity, so Z(D )={R } when n is odd.
n 0
Since the center will continue to either be of order 1 or order 2, no matter how
large the group is, the sizes of the centers may rapidly become an inaccurate
measure of the degree to which a dihedral group is non-Abelian.
Giventhatthecentersofthedihedralgroupsdonotprovideuswithanaccurate
method for measuring group commutativity,5 we proceed to compute the cen-
tralizersofeveryelementofeachofthefourdihedralgroupsdetailedabove. We
mayagainrefertotheCayleytableofeachofthesegroupsinordertodetermine
the centralizers of their elements according to Definition 0.5.
The elements of the centralizer of each element of D are listed below.
3
C(R )={R ,R ,R ,V,D,D(cid:48)},
0 0 120 240
C(R )={R ,R ,R },
120 0 120 240
C(R )={R ,R ,R },
240 0 120 240
C(V)={R ,V},
0
C(D)={R ,D},
0
C(D(cid:48))={R ,D(cid:48)}.
0
5Itshouldbenoted,however,thatthecenterofagroupwillbeanimportantconsideration
whendeterminingthemaximumcommutativityofnon-Abeliangroups.
7
The elements of the centralizer of each element of D are listed below.
4
C(R )={R ,R ,R ,R ,V,H,D,D(cid:48)},
0 0 90 180 270
C(R )={R ,R ,R ,R },
90 0 90 180 270
C(R )={R ,R ,R ,R ,V,H,D,D(cid:48)},
180 0 90 180 270
C(R )={R ,R ,R ,R },
270 0 90 180 270
C(V)={R ,R ,V,H,},
0 180
C(H)={R ,R ,V,H,},
0 180
C(D)={R ,R ,D,D(cid:48)},
0 180
C(D(cid:48))={R ,R ,D,D(cid:48)}.
0 180
The elements of the centralizer of each element of D are listed below.
5
C(R )={R ,R ,R ,R ,R ,V,D,D(cid:48),F,F(cid:48)},
0 0 72 144 216 288
C(R )={R ,R ,R ,R ,R },
72 0 72 144 216 288
C(R )={R ,R ,R ,R ,R },
144 0 72 144 216 288
C(R )={R ,R ,R ,R ,R },
216 0 72 144 216 288
C(R )={R ,R ,R ,R ,R },
288 0 72 144 216 288
C(V)={R ,V},
0
C(D)={R ,D},
0
C(D(cid:48))={R ,D(cid:48)},
0
C(F)={R ,F},
0
C(F(cid:48))={R ,F(cid:48)}.
0
The elements of the centralizer of each element of D are listed below.
6
C(R )={R ,R ,R ,R ,R ,R ,V,V(cid:48),V(cid:48)(cid:48),D,D(cid:48),D(cid:48)(cid:48)},
0 0 60 120 180 240 300
C(R )={R ,R ,R ,R ,R ,R },
60 0 60 120 180 240 300
C(R )={R ,R ,R ,R ,R },
120 0 60 120 180 240
C(R )={R ,R ,R ,R ,R ,R ,V,V(cid:48),V(cid:48)(cid:48),D,D(cid:48),D(cid:48)(cid:48)},
180 0 60 120 180 240 300
C(R )={R ,R ,R ,R ,R ,R },
240 0 60 120 180 240 300
C(R )={R ,R ,R ,R ,R ,R },
300 0 60 120 180 240 300
C(V)={R ,R ,V,D},
0 180
C(V(cid:48))={R ,R ,V(cid:48),D(cid:48)},
0 180
C(V(cid:48)(cid:48))={R ,R ,V(cid:48)(cid:48),D(cid:48)(cid:48)},
0 180
C(D)={R ,R ,V,D},
0 180
C(D(cid:48))={R ,R ,V(cid:48),D(cid:48)},
0 180
C(D(cid:48)(cid:48))={R ,R ,V(cid:48)(cid:48),D(cid:48)(cid:48)}.
0 180
8
As the order of the centralizer of a certain group element gives us the number
of group elements that commute with that certain element, we can rapidly
determine the total number of commutative pairs of elements (including the
product of an element with itself, which is clearly commutative) simply by
summing the orders of the centralizers of every element in a group. By this
standard, we can then also say that the total possible number of commutative
pairs of elements in a given group (that is, the total number of pairs) is simply
the order of the group squared.
Thus,thepreviouslymentionedP (G)measureofcommutativitycanbewritten
2
as the fraction c/|G|2, where c represents the sum of the the centralizers of all
of the group elements.
To better understand this definition of P (G), we will compute the P (D ) for
2 2 n
n=3,4,5,6. Aswehavealreadycomputedthecentralizersforeveryelementin
eachofthesefourdihedralgroups,itremainstosumtheorderofeachcentralizer
in a given group, and divide this value by the order of the group squared.
In this way, we find that in D , the total number of pairs that commute is
3
c = 6 + 3 + 3 + 2 + 2 + 2 = 18 (the sum of the order of each of the six
centralizers), while the order of the group squared is |D |2 = 62 = 36. Thus,
3
the degree to which D is commutative, is:
3
18/36=0.5
Likewise, for D , the number of commutative pairs is c = 8+4+8+4+4+
4
4+4+4=40, the total number of pairs is |D |2 =82 =64, and the degree to
4
which it is commutative is:
40/64=0.625
ForD ,thenumberofcommutativepairsisc=10+5+5+5+5+2+2+2+2+2=
5
40, the total number of pairs is |D |2 = 102 = 100, and the degree to which it
5
is commutative is:
40/100=0.4
Finally, for D , the number of commutative pairs is c = 12+6+6+12+6+
6
6+4+4+4+4+4+4=72, the total number of pairs is |D |2 =122 =144,
6
and the degree to which it is commutative is:
72/144=0.5
9
From these examples, we can begin to suppose that even-polygon dihedral
groupsaremorecommutativethansimilarly-sizedodd-polygondihedralgroups,
and that smaller dihedral groups are more commutative than larger ones. We
will shortly see that both of these ideas are correct, and can be easily proven.
Deriving P from conjugacy classes
2
As promised, we will now discuss a second method by which we may arrive at
the commutativity measure P . Since this method uses the idea of conjugacy
2
classes, let us begin with a definition.
Definition 1.2: Conjugacy Class of a.
Let a and b be elements of a group G. We say that a and b are conjugate in G
(and call b a conjugate of a) if xax−1 =b for some x in G. The conjugacy class
of a is the set cl(a)={xax−1 | x∈G}.
The reader may notice that conjugacy is an equivalence relation on a group.
That is, conjugacy satisfies the reflexive, symmetric, and transitive properties,
as shown below.
Let G be a group and let a ∈ G. Notice that a = eae−1, where e ∈ G is the
identityofthegroup. Thus, conjugacysatisfiesthereflexiveproperty. Next, let
b ∈ G and that a = xbx−1 for some x ∈ G. Then x−1ax = x−1xbx−1x = b,
such that conjugacy satisfies the symmetric property. Finally, let c ∈ G, and
suppose that in addition to a=xbx−1, there exists y ∈G such that b=ycy−1.
Thena=x(ycy−1)x−1 =(xy)c(xy)−1,whichshowsthatconjugacysatisfiesthe
transitive property.
Thisstraightforwardverificationthatconjugacysatisfiesthethreedefiningprop-
ertiesofanequivalencerelationallowsustohenceforthpartitionanygroupinto
disjoint conjugacy classes where necessary.
It is worth noting that proofs of several of the following theorems rely one one
ofthemostimportantresultsinfinitegrouptheory,Lagrange’sTheorem,which
is included here for reference.
Theorem 1.1. Lagrange’s Theorem: |H| divides |G|.
If G is a finite group and H is a subgroup of G, then |H| divides |G|. Moreover,
the number of distinct left cosets of H in G is |G|/|H|.
10