Most of the definitions and the results in this section are taken from [Rot65].
Basic Definitions. A group is an ordered pair(G;)whereGis a set andis a binary operation
onGsatisfying the following properties:
g1 g 1 (g 2 g 3 )=(g 1 g 2 )g 3for all g 1 ;g 2 ;g 3 2G.
g2 There existsid2G(the identity) such thatgid=g=idgfor allg2G.
g3 For allg 1 2Gthere existsg 2 2G(the inverse ofg 1, often denoted by g 1 1 ) such thatg 1 g 2 = id=g 2 g 1.
IfXis a nonempty set, a permutation ofX is a bijective functiongonX. LetS
X denote
the set of permutations onX. Although most of the definitions are general, in all our subsequent
discussionXwill be the set[n℄.
There are many ways to represent permutations. We will normally use the cycle notation defined as follows:
(2) Join the point labelledito the point labelledjby an edge with an arrow
pointing towardsjifg(i)=j. (This will form a number of cycles).
(3) Write down a list(i 1
;i 2
;:::;i k
)for each cycle formed in step (2).
(4) Remove all lists formed by a single element. So for exampleg 2 S
6 with
g(1) = 3, g(2) = 2,g(3) = 4g(4) = 1,g(5) = 6and g(6)=5will be represented as(134)(56). It is possible to associate a unique multisetf1:k
1 ;2: k 2 ;:::;n:k n
g(sometimes represented symbolically asx k 1 1 x k 2 2 :::x kn n or simply[k 1 ;k 2 ;:::;k n ℄) to every permutationg2S
ndescribing its cycle structure (or cycle type): ghask
icycles of length i. In particulark
1is the number of elements of
[n℄that are fixed byg, i.e. such thatg(i) =i. If g(i)6=iwe say thatgmovesi.
Ifg 1 ;g 2 2S Xthen g 1 Æg 2is a new function on Xsuch that(g 1 Æg 2 )(x)= df g 1 (g 2 (x)). It
is easy to verify thatg 1 Æg 2 2S X. The pair (S X
;Æ)is indeed a group called the symmetric group
onX.S
nwill denote both the set S
[n℄and the group (S
[n℄ ;Æ).
Subgroups and Lagrange Theorem. If(G;)is a group, a nonempty subsetHofGis a subgroup
of(G;)if sg1 g 1 g 2 2Hfor allg 1 ;g 2 2H.
sg2 The identity of(G;)belongs toH.
sg3 g
1
2Hfor allg2H.
Theorem 12 IfH is a subgroup of a groupGthen there existsm2IN +
such thatjGj=mjHj.
Proof. (Sketch, see [Rot65] for details) GivenHandg2G, define the setgH =fgh:h2Hg.
It follows from g1-g3 and sg1-sg3 that
1. jgHj=jHjfor allg2G. 2. Ifg 1 6=g 2 2Gthen eitherg 1 H =g 2 H org 1 H\g 2 H =;. 3. For allg 1 2Gthere existsg 2 2Gsuch thatg 1 2g 2 H.
So there exists anmsuch thatG=g 1
H[:::[g m
H and the setsg i
H form a partition ofG. 2
The study of permutation groups is strictly related to the study of graphs because a graph provides a picture of a particular type of subgroup inS
Definition 2 [HP73] Given a graphG =(V;E)the collection of all permutationsg 2 S V such
thatfu;vg2 Eif and only iffg(u);g(v)g 2Efor allu;v 2V is the automorphism group ofG
and is denoted by Aut(G).
The structure and the properties of the automorphism group of a graph are of particular importance in the study of unlabelled graphs and isomorphisms between labelled graphs.
Action Theory. A group(G;)acts on a setif there is a function (called action):G!
such that 1. id=for each2. 2. g 1 (g 2 )=(g 1 g 2 )for allg 1 ;g 2 2Gand2.
The action ofGoninduces an equivalence relationon(if and only if=gfor some g2G). The equivalence classes are called orbits. For eachg2G, defineFix(g)=f2:g= gand conversely for each2define the stabilizer ofto be the set
=fg2G:g=g.
Lemma 1 is a subgroup of
G.
In particularS
ncan be acting on itself:
fg=fÆgÆf 1
. In this caseis called conjugacy
relation and the orbits are called conjugacy classes. In what followsCwill denote a conjugacy class
inS n.
Theorem 13 Conjugacy classes inS
nare formed by all permutations with the same cycle type.
Proof. Ifg =:::(:::ij:::):::thenhÆgÆh 1
has the same effect of applyinghto the elements
ofghencegandhÆgÆh 1
have the same cycle type. Letf andgbelong to the same conjugacy
class. Thenf =hÆgÆh 1
for someh2S
n. But this implies that
fhas the same cycle type ofg.
Conversely iffandghave the same cycle type, align the two cycle notations, definehand
it is easy to prove thatf andgare conjugate. 2
Thus the number of different conjugacy classes is the same as the number of different cycles types. From now on, a conjugacy classCwill be identified with the decomposition ofndefining the
cycle type of the permutations inC. The following result is well known (see for example [Kri86]).
Theorem 14 The number of permutations with cycle type[k
1 ;:::;k n ℄is n! Q n i=1 (i k ik i !) .
Proof. Given the form of the cycle notation ( )( ):::( ) | {z } k 1 ( ; )( ; ):::( ; ) | {z } k 2 :::( ; ;:::; ) | {z } x | {z } k x
it is possible to count the number of ways to fill it.
There aren!ways to fill thenplaces. The firstk
1unary cycles can be arranged in k
1 !ways. Thek
2 cycles of length 2 can be arranged in k
2
! ways times for each of thek
2 cycles the
possible ways to start (two). Sok 2 !2 k 2 overall. Similarly fork i, there are
iways to start one of thei-cycles. Hencek i
!i ki
ways to putik i
chosen items in cycles of lengthi.
2
The following theorem states a couple of well known results which will be useful.
Theorem 15 LetGbe a finite group acting on a set6=;.
1. (Orbit-Stabilizer Theorem) For each orbit!,jf(g;):2!\Fix(g)gj=jGj.
2. (Fr¨obenius-Burnside Lemma) The number of orbits is
m= 1 jGj X 2 j j= 1 jGj X g2G jFix(g)j:
Proof. For each orbit! the elements off(g;) : 2 !\Fix(g)g are pairs withg 2
and 2!. There arej
jj!jof these pairs. The first result follows from Theorem 12 applied to
since there is a bijectionbetween! and the collectiong i : if
2 !then =gfor some g2G; define()=g
.
The first part of the second result follows from the first result. Assume there are! 1
;:::;! m
different orbits. Summing over all2!
iwe have X 2!i j! i jj j= X 2!i jGj
and from this
X 2! i j! i jj j=j! i jjGj
and finally, simplifying on both sides X 2!i j j=jGj
Finally, adding over all orbits
m X i=1 X 2!i j j=mjGj
To understand the second equality observe that the sum on the left in the expression above is counting pairs(g;)for 2 andg 2
. This is equivalent to count pairs
(g;) forg 2 G and 2 Fix(G). Hence m X i=1 X 2! i j j= X g2G jFix(g)j 2
Pair Group and Combinatorial Results. LetS
(2)
n be the permutation group on the set of un-
ordered pairs of numbers in[n℄. Every permutationg2S
ninduces a permutation g 2S (2) n defined byg (fi;jg)=fg(i);g(j)g. Theorem 16 Letf;g2CS
nand assume the cycle type of Cis[k 1 ;:::;k n ℄. Then 1. f g ; 2. jFix(g)j=2 q(C)
whereq(C)is the number of cycles ofg 2S (2) n definable in terms of g; 3. If'(n) = df
jfx : 1 x < n;gd(n;x) = 1gjis the Euler totient function andl(i) = P dn=ie j=1 k ij then q(C)= 1 2 ( n X i=1 l(i) 2 '(i) l(1)+l(2) )
4. jFix(f)\!j=jFix(g)\!jfor every orbit!.
Proof. For everyg 2S
nthe cycle type of g
only depends on the cycle type ofg(see for example
[HP73, p. 84]). The first statement is then immediate. The second statement follows from Theorem 15.2 and the formula for the number of unlabelled graphs given by the P´olya enumeration theorem (see [HP73, Section 4.1]). The third and fourth results are mentioned in [DW83]. 2