Symmetric Group Sym(n)

• As we know a permutation p is a bijective mapping of a set A onto itself: p : A  A. Permutations may be multiplied and form the symmetric group Sym(A) = Sym(n) = S n = S A , that has n! elements, where n = |A|.

Permutation Group

• Any subgroup G · Sym(A) is called a

permutation group

. If we consider an abstract group G then we say that G

acts on

A.

• In general the

group action

( G , A, f ), where G is defined as a triple is a group, A a set and f : G !

Sym(A) a group homomorphism.

• In general we are only interested in

faithful actions

, i.e. actions in which f is an isomorphism between G and f ( G ).

Automorphisms of Simple Graphs

• Let X be a simple graph. A permutation h:V(X) !

V(X) is called an

automorphism

of graph X if for any pair of vertices x,y 2 V(X) x~y if and only if h(x)~h(y). By Aut X we denote the group of automorphisms of X.

• Aut X is a permutation group, since it is a subgroup of Sym(V(X)).

Orbits and Transitive Action

• Let G be a permutation group acting on A and x 2 A. The set [x] := {g(x)|g 2 G} is called the G[x] = [x].

orbit

of x. We may also write • G defines a partition of A into orbits: A = [x 1 ] t [x 2 ] t ... t [x k ].

• G

acts transitively

on A if it induces a single orbit.

Example

• Aut G(6,2) induces two orbits on the vertex set.

• Aut G(6,2) induces an action on the edge set. There we get three orbits.

Orbits

• Let G acts on space V. On V an equivalecne relation ¼ is introduces as follows: • x ¼ y , 9 a 2 G 3 : y = a (x).

• Equivalence, indeed: » Reflexive » Symmetric » Transitive • [x] ... Equivalence class to with x belongs is called an

orbit

. (Also denoted by G [x].)

1 d 3 a e c

Example

2 4 b • Graph G=(V,E) has four automorphisms.

• V(G) ={1,2,3,4} splits into two orbits [1] = {1,4} and [2] = {2,3}.

• E(G) = {a,b,c,d,e} also splits into two orbits: [a] = {a,b,e,d} and [c] = {c}.

Homewrok

• H1. Let X be any of the three graphs below.

• Determine the (abstract) group of automorphisms Aut X.

• Action of Aut X on V(X).

• Action of Aut X on E(X).

X 1 X 2 X 3

Stabilizers and Orbits

• Let G be a permutation group acting on A and let x 2 A. By G(x) we denote the

orbit

of x. • G(x) = {y 2 A| 9 g 2 G 3 : g(x) = y} • Let G x µ G be the set of group elements, fixing x. G x is called the

stabilizer

of x and forms a subgroup of G.

Orbit-Stabilizer Theorem

• •

Theorem:

|G(x)||G x | = |G|.

Corollary

: If G acts transitively on A then |A| is the index of any stabilizer G x in G.

Burnside’s Lemma

• Let G be a group acting on A.

• For g 2 G let fix(g) denote the number of fixed points of permutation g. • Let N be the number of orbits of G on A.

• Then:

Regular Actions

• The transitive action of G on A is called

regular

, if |G| = |A|, or equivalently, if each stabilizier is trivial.

• An important and interesting question can be asked for any transtive action of G on A.

• Does G have a subgroup H acting regularly on A?

Semiregular Action

•

Definition:

Grup G acts on V

semiregulary

, • If there exists a 2 G 3 : a = ( ...) ( ...) ...( ...) composed of cycles of the same size r; |V| = r s.

• For each x 2 V we have: |[x]| = r.

Primitive Groups

• A transitive action of G on X is called

imprimitive

, if X can be partitioned into k (1 < k < |X|) sets: X = X 1 t X 2 t ... t X k (called

blocks of imprimitivity

)and each g 2 G induces a set-wise permutation of the X i ’s.

• If a group is not imprimitive, it is called

primitive

.

Example

• For a prism graph P n , Aut P n if and only if n  4.

is imprimitive • There are n blocks of imprimitivity of size 2, each corresponding to two endpoints of a side edge.

Permutation Matrices

• Each permutation p 2 Sym(n) gives rise to a permutation matrix P( p ) = [p ij ] with p ij = 1 if j = p (i) and p ij = 0 otherwise.

• Example: p 1 = [2,3,4,5,1] and P( p 1 ) is shown below: 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0

Matrix Representation

• A permutation group G can be represented by permutation matrices. There is an isomorphism p a P( p ). And p s correspons to P( p )P( s ). Since each permutation matrix is orthogonal, we have P( p -1 ) = P t ( p ).

Alternating Group Alt(n)

• A

transposition

t is a permutation interchanging a single pair of elements.

• Permutation p is

even

if it can be written as a product of an even number of transpositions (otherwise it is

odd

.) • Even permutations from Sym(n) form the

alternating group

Alt(n), a subgroup of index 2.

Iso(M)

• Isometries of a metric space (M,d) onto itslef form a

group of isometries

denote by Iso(M).

that we

Sim

1

(M)

• Similarities of a metric space (M,d) onto itslef form a

group of similarities

denote by Sim 1 (M).

that we

Sim

2

(M)

• Similarities of a metric space (M,d) onto itslef form a

group of similarities

that we denote by Sim 2 (M).

• In any metrc space the groups are related: • Iso(M) · Sim 2 (M) · Sim 1 (M).

Symmetry

• • Let X µ M be a set in a metric space (M,d). An isometry s 2 Iso(M) that fixes X set-wise: s (X) = X, is called a

(metric) symmetry

of X.

• All symmetries of X form a group that we denote by Iso M (X) or just I(X). It is called the

symmetry group

of X.

Note:

this idea can be generalized to other groups and to other structures!

Free Group F(

S

)

• Let S be a finite non-empty set. Form two copies of it, call the first S + , and the second S . Take all words ( S + t S ) * over the alphabet S + Introduce an equivalence relation @ that two words u @ t S . in such a way v if and only if one can be obtained from the other one by a finite series of deletion or insertion of adjacent a + a or a a + .

• Let F( S ) = ( S + t S ) * / @ . Then F( S ) is a group, called the free group generated by S .

• We also denote F( S ) = < S | >.

Finitely Presented Groups

• Let S R k } ½ and < S ( S + t | > be as before. Let R = {R 1 , R 2 , ..., S ) * be a set of

relators

.

• The expression < S | R> is called a

group presentation

. It defines a quotient group of < S | >. • Two group elements from F( S ) are equivalent if one can be obtained from the other by insertion or delition of the relators R and their inverses.

• Since both sets \Sigma and R are finite, the group is

finitely presented

.

Generators

• Let G be a group and X ½ X = X 1 and 1  G. Assume that X. Then X is called the set of generators. Let denote the smallest subgroup of G that contains X. We say that X generates .

Cayley Theorem

• •

Theorem.

Every group G is isomorphic to some permutation group.

Proof

. For g G by x a 2 G define its xg. The mapping from G to Sym(G) defind by g a (x a isomorphism to its image.

right action

xg) is an on

Cyclic Group Cyc(n)

• Let G = . Hence G = {1,a,a 2 ,..,a n-1 }. By Calyey Theorem we may represent a as the cyclic permutation (2,3,...,n,1) that generate the group Cyc(n) · Sym(n).

• Note that Cyc(n) is isomorphic to ( Z n ,+). • Cyc(n) may also be considered as a symmetry group of some polygons. Cyc(8) is the symmetry group of the polygon on the left.

3 2 4

Dihedral Group Dih(n)

t 1 5 s 6 • Dihedral group Dih(n) of order 2n is isomoprihc to the symmetry group of a regular n-gon.

• For instance, for n=6 we can generate it by two permutations: s = (2,3,4,5,6,1) and t = (1,2)(3,6)(4,5). Dih(n) has the following presentation: •

Symmetry of Platnoic Solids

• There are five Platonic solids: Tetrahedron T, Octahedron O, Hexaedron H, Dodecahedron D and Icosahedron I.

Tetrahedron

• Tetrahedron has • v = 4 vertices, • e = 6 edges and • f = 4 faces.

• Determine its symmetry group.

Octahedron

• Octahedron has • v = 6 vertices, • e = 12 edges and • f = 8 faces.

• Determine its symmetry group

Hexahedron

• Hexahedron has • v = 8 vertices • e = 12 edges and • f = 6 faces.

• Determine its symmetry group

Dodecahedron

• Dodecahedron has • v = 20 vertices, • e = 30 edges and • f = 12 faces.

• Determine its symmetry group

Icosahedron

• Icosahedron has • v = 12 vertices, • e = 30 edges and • f = 20 faces.

• Determine its symmetry group

Skeleton of Tetrahedron – T

S

= K

4 • K 4 has • v = 4 vertices, • e = 6 edges • f = 4 triangles.

• Aut(K 4 ) = S 4 .

Skeleton of Octahedron – O

S

= K

2,2,2 • O S has • v = 6 vertices, • e = 12 edges

Skeleton of Hexahedron H

S

=K

2 ¤

K

2 ¤

K

2 • H S ima • v = 8 vertices • e = 12 edges

Skeleton of Dodecahedron D

S

= G(10,2)

• G(10,2) has • v = 20 vertices, • e = 30 edges

Skeleton of Icosahedron I

S • It has • v = 12 vertices, • e = 30 edges

Platonic Solids and Symmetry

• We only considered the groups of direct symmetries (orientation preserving isometries). • The full group of isometries coincides (in this case) with the group of automorphisms of the corresponding graphs.

• In general: • Sym + (M) · Sym(M) · Aut(M S ).

Homework

• H1. Determine the group of symmetries of the prism P 6 .

• H2. Determine the group of symmteries of the antiprism A 6 .

• H3. Determine the group of automorphism for the pyramid P 6 .

• H4. Determine the group of symmetries of the double pyramid B of n.

skeleta.

6 .

• H5. Generalize for other values • H6. Repeat the problems for the