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 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).

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 = [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 ,9a2G3: 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 Gxµ 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. 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 G acts on V semiregulary, • If there exists a2G3: 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 < k < |X|) sets: X = X1t X2t ... 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.

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 p2 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:

17. Matrix Representation • A permutation group G can be represented by permutation matrices. There is an isomorphism pa P(p). And ps 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 transpositiont 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 s2 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+tS-)* over the alphabet S+tS-. 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(S) = (S+tS-)* / @ . Then F(S) is a group, called the free group generated by S. • We also denote F(S) = <S | >.

24. Finitely Presented Groups • Let S and <S | > be as before. Let R = {R1, R2, ..., Rk} ½ (S+tS-)* 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.

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

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 = <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.

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 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: • <s,t|sn=t2=stst=1> 2 1 3 6 s 4 5

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.