Draft of September 8, 2017 GRUPOS NANTEL BERGERON Abstract. 1. Quick Introduction In this mini course we will see how to count several attribute related to symmetries of an object. For example, how many different dices with four colors can we construct? . ??? There are many possibilities and it is hard to answer without the tools we will build. Another typical example is how many different necklaces can we build using 3 kind of beads? • • • • • • • • • • • • • • • • • • • • • • • • • . ?? • • • • • • • • • • • • • • • To achieve this we need to learn about groups of symmetries, action of groups, cosets, etc We will see a much more interesting enumerations and applications. 2. Simetrias y groupos We are interested in understanding all the symmetries of an object we study and the algebraic structures on those symmetries. 2.1. Conjuntos de Simetrias. Suppose you have a solid square attached in the middle: • • • • • We can turn the square and get back the same thing. This is a symmetry. Research of Bergeron supported in part by NSERC and York Research Chair. 1 2 BERGERON • • • • • We want to list all the possibilities: • • • • • • • • • • • • • • • • • • • • Is there more? When are two symmetries considered the same?? To keep track of what happen, We can name the vertices of the square to see what happen to them. This is a secret marking, it is not part of the object, it is only there to help us keep track of what happen: d• •c c• •b b• •a a• •d • • • • • • • • • • • • a b d a c d b c We can then encode what happen to the square by only listing what happen to a; b; c; and d: abcd abcd abcd abcd σ = ; σ = ; σ = ; σ = : 0 abcd 1 dabc 2 cdab 3 bcda This is a mathematical model of the symmetries of the square. Now we say that two symme- tries are equal if starting with the same marking, they end up with the same marking after GRUPOS 3 the symmetry. d• •c c• •b a• •d • 7! • 7! • • • • • • • a b d a b c d• •c a• •d a• •d • 7! • = • • • • • • • a b b c b c This model is useful to understand the structures of those symmetries and focus on what interests us at the moment (the vertices at the corner of the square). This describe in some way all the symmetries of the square. I think of these as maps. This is not unique, we will consider soon other descriptions of the same symmetries and see that they can have different presentations. 2.2. operations on symmetries. We now want to describe operation we can do on sym- metries. First, composing symmetries: What happen if I do a rotation, and then another one from what we have done? d• •c c• •b b• •a b• •a • 7! • 7! • = • • • • • • • • • a b d a c d c d Now using our mathematical model: a b c d #### abcd d a b c = : #### cdab c d a b I think of these as composition of maps: σ1 : fa; b; c; dg ! fa; b; c; dg σ2 : fa; b; c; dg ! fa; b; c; dg a 7! d a 7! c b 7! a b 7! d c 7! b c 7! a d 7! c d 7! b What we have seen above is that σ1 ◦ σ1 = σ2 4 BERGERON it is the same maps. Another example d• •c a• •d c• •b c• •b • 7! • 7! • = • • • • • • • • • a b b c d a d a In all case composing rotations gives us another rotation. Now using our mathematical model we can look at all cases abcd abcd abcd abcd abcd dabc cdab bcda abcd abcd abcd abcd abcd abcd abcd dabc cdab bcda abcd abcd abcd abcd abcd dabc dabc cdab bcda abcd abcd abcd abcd abcd abcd cdab cdab bcda abcd dabc abcd abcd abcd abcd abcd bcda bcda abcd dabc cdab The identity symmetry: You remarked that there is a special symmetry that \does noth- ing": d• •c d• •c abcd • 7! • encoded by Id = abcd • • • • a b a b This symmetry is special in the sense that if we compose by Id on the right or on the left we do not change anything: Id ◦ σ = σ ◦ Id = σ Inverting symmetries: Finally, you see that any symmetry can be undone: d• •c a• •d d• •c • 7! • 7! • = Id • • • • • • a b b c a b GRUPOS 5 Here we have that d• •c c• •b c• •b • 7! • = • • • • • • • a b d a d a We say that it is the inverse. If σ is a symmetry, we denote its inverse by σ−1. Using notation as above, we have −1 −1 −1 −1 σ0 = σ0 σ1 = σ3 σ2 = σ2 σ3 = σ1 2.3. Abstract groups. When we talk about the symmetries of an object we have seen four important features (1) We have a set of symmetries (2) We can compose symmetries and it gives back a symmetry in our set (3) There is a special symmetry called the identity. (4) Every symmetry has an inverse In mathematical terms, we have a group. More formally, a group is: (1)A set G (2) An associative operation m: G × G ! G [we often write m(a; b) = ab or m(a; b) = a + b] associative: a(bc) = (ab)c (3) A unique element 1 2 G: for all g 2 G 1g = g1 = g (4) Every g 2 G has an inverse g−1: gg−1 = g−1g = 1 2 3 4 Example 1. C4 = f1; a; a ; a g where we assume a = 1 so m 1 a a2 a3 1 1 a a2 a3 a a a2 a3 1 a2 a2 a3 1 a a3 a3 1 a a2 It is associative. We have the special element 1 and every element has an inverse a−1 = a3; a−2 = a2; a−3 = a: 6 BERGERON 2 2 Example 2. C2 × C2 = f1; a; b; abg where we assume a = 1, b = 1 and ab = ba so m 1 a b ab 1 1 a b ab a a 1 ab b b b ab 1 a ab ab b a 1 It is associative. We have the special element 1 and every element has an inverse a−1 = a; b−1 = b; (ab)−1 = ab: 2 2 Example 3. S3 = f1; s; t; st; ts; stsg where we assume s = 1, t = 1 and sts = tst (is it all?), so m 1 s t st ts sts 1 1 s t st ts sts s s 1 st t sts ts t t ts 1 tst s st st st sts s ts 1 t ts ts t sts 1 st s sts sts st ts s t 1 It is associative. We have the special element 1 and every element has an inverse a−1 = a; b−1 = b; (ab)−1 = ab: 2.4. Symmetric group. We have seen above that symmetries of an object can be encoded by bijective maps on a finite set. This raises two questions { Is it possible to realize any (abstract) group with bijective group (equivalently, is it true that abstract groups are the symmetries of something?) { Is the set of all bijections of a finite set a group? Let us first answer the second question. Let [n] = f1; 2; : : : ; ng and consider the set Sn = fσjσ :[n] ! [n] bijectiong We say that the element of Sn are permutations. For example S3 = f123; 132; 213; 231; 312; 321g where we encoded the permutations by its list of values. That is using the natural order 1 < 2 < : : : < n, we list the values σ(1); σ(2); : : : ; σ(n). For example the permutation 231 is the map f1; 2; 3g ! f1; 2; 3g 1 7! 2 2 7! 3 3 7! 1 We have the identity permutation Id = 123 ··· n 2 Sn and given two permutations σ; π 2 Sn we can compose the two maps and we get a permutation σ ◦ π 2 Sn. Moreover, for every −1 permutation σ 2 Sn we can find σ such that σ ◦ σ−1 = σ−1 ◦ σ = Id: GRUPOS 7 This gives us a nice group and we call it the symmetric group. 2.5. Representation by permutation. Let us now consider the question of seeing an abstract group as a group of symmetry. Look at Example 1. You can check that the map 1 7! 1234; a 7! 2341; a2 7! 3412; a3 7! 4123: is a realization of the group C4. Now compare this with our starting example of rotating the square and see that up to changing the names, we have the same group of symmetries. The abstract group C4 is realized using permutation (a subset of S4). f1234; 2341; 3412; 4123g ⊂ S4: We need to make some observations. When we have fIdg ⊆ H ⊆ Sn and H is a group by itself we say that H is a permutation subgroup of Sn. For a group G, if we have a map ': G ! Sn such that '(1) = Id and '(ab) = '(a) ◦ '(b), then we say that ' is a homomorphism of G. You can check that in this case fIdg ⊆ '(G) ⊆ Sn is then a permutation subgroup of Sn and we say that '(G) is a permutation representation of G. If moreover jGj = j'(G)j then we say that '(G) is a permutation realization of G. Exercises. Ex.2.1 For C4 in Example 1, find a homomorphism ': C4 ! S5. Find 5 points on the square that are maps to themselves after rotations and visualize your construction of '. Ex.2.2 Put numbers on the vertices and the sides of the square 4• 7 •3 5 • 6 • • 1 8 2 use this to define a homomorphism φ: C4 ! S8.