Symmetric Group Sym(n)

• As we know a permutation is a bijective mapping of a set A onto itself: : A A. Permutations may be multiplied and form the symmetric group Sym(A) = Sym(n) = Sn = SA, 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 is defined as a triple (, A, ), where is a group, A a set and : ! Sym(A) a group homomorphism.

• In general we are only interested in faithful actions, i.e. actions in which is an isomorphism between and ().

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 orbit of x. We may also write G[x] = [x].

• G defines a partition of A into orbits: A = [x1] t [x2] t ... t [xk].

• 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 acts on space V. On V an equivalecne relation ¼ is introduces as follows:

• x ¼ y , 9 2 3: y = (x).

• Equivalence, indeed:» Reflexive

» Symmetric

» Transitive

• [x] ... Equivalence class to with x belongs is called an orbit. (Also denoted by [x].)

Example

• 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}.

1

3 4

2a

e

c bd

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).

X1X2

X3

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 Gx µ G be the set of group elements, fixing x. Gx is called the stabilizer of x and forms a subgroup of G.

Orbit-Stabilizer Theorem

• Theorem: |G(x)||Gx| = |G|.

• Corollary: If G acts transitively on A then |A| is the index of any stabilizer Gx 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 acts on V semiregulary,

• If there exists 2 3: = ( ...) ( ...) ...( ...) 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 = X1 t X2 t ... t Xk (called blocks of imprimitivity)and each g 2 G induces a set-wise permutation of the Xi’s.

• If a group is not imprimitive, it is called primitive.

Example

• For a prism graph n, Aut n is imprimitive if and only if n 4.

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

Permutation Matrices

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

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

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

1 0 0 0 0

Matrix Representation

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

Alternating Group Alt(n)

• A transposition is a permutation interchanging a single pair of elements.

• Permutation 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 that we denote by Iso(M).

Sim1(M)

• Similarities of a metric space (M,d) onto itslef form a group of similarities that we denote by Sim1(M).

Sim2(M)

• Similarities of a metric space (M,d) onto itslef form a group of similarities that we denote by Sim2(M).

• In any metrc space the groups are related:

• Iso(M) · Sim2(M) · Sim1(M).

Symmetry

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

• All symmetries of X form a group that we denote by IsoM(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()

• Let be a finite non-empty set. Form two copies of it, call the first +, and the second -. Take all words (+ t -)* over the alphabet + t -. Introduce an equivalence relation in such a way that two words u 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() = (+ t -)* / . Then F() is a group, called the free group generated by .

• We also denote F() = < | >.

Finitely Presented Groups

• Let and < | > be as before. Let R = {R1, R2, ..., Rk} ½ (+ t -)* be a set of relators.

• The expression < | R> is called a group presentation. It defines a quotient group of < | >.

• Two group elements from F() 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 G. Assume that X = X- and 1 X. Then X is called the set of generators. Let <X> denote the smallest subgroup of G that contains X. We say that X generates <X>.

Cayley Theorem

• Theorem. Every group G is isomorphic to some permutation group.

• Proof. For g 2 G define its right action on G by x xg. The mapping from G to Sym(G) defind by g (x xg) is an isomorphism to its image.

Cyclic Group Cyc(n)

• Let G = <a| an>. Hence G = {1,a,a2,..,an-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 (Zn,+).

• 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.

Dihedral Group Dih(n)

• 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: = (2,3,4,5,6,1) and = (1,2)(3,6)(4,5). Dih(n) has the following presentation:

• <s,t|sn=t2=stst=1>

54

3

2 1

6

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 – TS = K4

• K4 has

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

• Aut(K4) = S4.

Skeleton of Octahedron – OS = K2,2,2

• OS has • v = 6 vertices, • e = 12 edges

Skeleton of Hexahedron HS =K2 ¤ K2 ¤ K2

• HS ima • v = 8 vertices • e = 12 edges

Skeleton of Dodecahedron DS = G(10,2)

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

Skeleton of Icosahedron IS

• 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(MS).

Homework

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

• H2. Determine the group of symmteries of the antiprism A6.

• H3. Determine the group of automorphism for the pyramid P6.

• H4. Determine the group of symmetries of the double pyramid B6.

• H5. Generalize for other values of n.

• H6. Repeat the problems for the skeleta.