Symmetric Group Sym(n)

1
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) Sn SA, that has n! elements,
  where n A.

2
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 (G, A, f), where G is a group, A a set
  and fG ! 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).

3
Automorphisms of Simple Graphs

• Let X be a simple graph. A permutation hV(X) !
  V(X) is called an automorphism of graph X if for
  any pair of vertices x,y 2 V(X) xy 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)).

4
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 Gx 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.

5
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.

6
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 ?x.)

7
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.

a
1
2
c
b
d
e
3
4
8
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).

X3
X2
X1
9
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.

10
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.

11
Burnsides 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

12
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?

13
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.

14
Primitive Groups

• A transitive action of G on X is called
  imprimitive, if X can be partitioned into k (1 lt
  k lt 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 Xis.
• If a group is not imprimitive, it is called
  primitive.

15
Example

• For a prism graph Pn, Aut Pn is imprimitive if
  and only if n ¹ 4.
• There are n blocks of imprimitivity of size 2,
  each corresponding to two endpoints of a side
  edge.

16
Permutation Matrices

• Each permutation p 2 Sym(n) gives rise to a
  permutation matrix P(p) pij with pij 1 if j
  p(i) and pij 0 otherwise.
• Example p1 2,3,4,5,1 and P(p1) 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
17
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)
  Pt(p).

18
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.

19
Iso(M)

• Isometries of a metric space (M,d) onto itslef
  form a group of isometries that we denote by
  Iso(M).

20
Sim1(M)

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

21
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).

22
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 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!

23
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 t S-.
  Introduce an equivalence relation _at_ in such a way
  that two words u _at_ v if and only if one can be
  obtained from the other one by a finite series of
  deletion or insertion of adjacent aa- or a-a.
• Let F(S) (S t S-) / _at_ . Then F(S) is a group,
  called the free group generated by S.
• We also denote F(S) ltS gt.

24
Finitely Presented Groups

• Let S and ltS gt be as before. Let R R1, R2,
  ..., Rk ½ (S t S-) be a set of relators.
• The expression ltS Rgt is called a group
  presentation. It defines a quotient group of ltS
  gt.
• 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.

25
Generators

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

26
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 a xg. The mapping from G to Sym(G) defind by g
  ? (x ? xg) is an isomorphism to its image.

27
Cyclic Group Cyc(n)

• Let G lta angt. 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.

28
Dihedral Group Dih(n)
t

• Dihedral group Dih(n) of order 2n is isomoprihc
  to the symmetry group of a regular n-gon.
• For instance, for n6 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
• lts,tsnt2stst1gt

2
1
3
6
s
5
4
29
Symmetry of Platnoic Solids

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

30
Tetrahedron

• Tetrahedron has
• v 4 vertices,
• e 6 edges and
• f 4 faces.
• Determine its symmetry group.

31
Octahedron

• Octahedron has
• v 6 vertices,
• e 12 edges and
• f 8 faces.
• Determine its symmetry group

32
Hexahedron

• Hexahedron has
• v 8 vertices
• e 12 edges and
• f 6 faces.
• Determine its symmetry group

33
Dodecahedron

• Dodecahedron has
• v 20 vertices,
• e 30 edges and
• f 12 faces.
• Determine its symmetry group

34
Icosahedron

• Icosahedron has
• v 12 vertices,
• e 30 edges and
• f 20 faces.
• Determine its symmetry group

35
Skeleton of Tetrahedron TS K4

• K4 has
• v 4 vertices,
• e 6 edges
• f 4 triangles.
• Aut(K4) S4.

36
Skeleton of Octahedron OS K2,2,2

• OS has
• v 6 vertices,
• e 12 edges

37
Skeleton of Hexahedron HS K2 K2 K2

• HS ima
• v 8 vertices
• e 12 edges

38
Skeleton of Dodecahedron DS G(10,2)

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

39
Skeleton of Icosahedron IS

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

40
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).

41
Homework

• H1. Determine the group of symmetries of the
  prism P6.
• 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.