1. 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) = S n = S A, 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 ( , 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  (  ).

3. 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)).

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 G[x] = [x]. 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.

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  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].)

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}. 1 34 2 a e cb d

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). X1X1 X2X2 X3X3

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 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.

10. 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.

11. 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:

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  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.

14. 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.

15. 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.

16. Permutation Matrices Each permutation  2 Sym(n) gives rise to a permutation matrix P(  ) = [p ij ] with p ij = 1 if j =  (i) and p ij = 0 otherwise. Example:  1 = [2,3,4,5,1] and P(  1 ) is shown below: 01000 00100 00010 00001 10000

17. 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 ) = P t (  ).

18. 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.

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

20. Sim 1 (M) Similarities of a metric space (M,d) onto itslef form a group of similarities that we denote by Sim 1 (M).

21. 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).

22. 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 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!

23. 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(  ) =.

24. Finitely Presented Groups Let  and be as before. Let R = {R 1, R 2,..., R k } ½ (  + t  - ) * be a set of relators. The expression 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.

25. 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 denote the smallest subgroup of G that contains X. We say that X generates.

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  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 =. 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.

28. 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:   54 3 21 6

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 – T S = K 4 K 4 has v = 4 vertices, e = 6 edges f = 4 triangles. Aut(K 4 ) = S 4.

36. Skeleton of Octahedron – O S = K 2,2,2 O S has v = 6 vertices, e = 12 edges

37. Skeleton of Hexahedron H S =K 2 ¤ K 2 ¤ K 2 H S ima v = 8 vertices e = 12 edges

38. Skeleton of Dodecahedron D S = G(10,2) G(10,2) has v = 20 vertices, e = 30 edges

39. Skeleton of Icosahedron I S 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(M S ).

41. Homework H1. Determine the group of symmetries of the prism  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 6. H5. Generalize for other values of n. H6. Repeat the problems for the skeleta.