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1 r r r r r s rs r s r s r s r s Warmup: Draw the symmetries for the triangle. () How many symmetries are there? () If we call the move rotate clockwise r, what is the order of r? Is there a way to write r in terms of some positive power of r? () If we call the move flip across a vertical axis s, what is the order of s? Isthereawaytowrites in terms of some positive power of s? () Note that r a s b means flip b times and then rotate a times (read actions right to left like function composition). Now label each of the symmetries by some r a s b. Then label each of the symmetries by some s b r a,andthenbys b r a, and compare all three forms. Repeat parts () () for the square.

2 Review: Let G be a set. A binary operation? on G is a function? : G G! G. A group is a pair (G,?) consisting of a set G and a binary operation? on G such that:.? is associative, i.e.(a? b)? c = a? (b? c);. there is an identity element e G, i.e. e? g = g = g? e for all g G;. every element of G has an inverse; i.e.forallg G, thereis an element g such that gg = e = g g. Favorite examples so far:. Z n, Q n, R n, C n under addition.. Q, R, C under multiplication.. Z/nZ under addition.. (Z/nZ) = {a Z/nZ a is relatively prime to n} under multiplication. Order The order of a group G, denoted G, is the size of the underlying set. For any element x G, ifx n = e for some n Z >0,wesaythe order of x is the smallest such n. Theorem. An element x G has order if and only if x = e.. x m = e i x divides m.

3 r r r r r s rs r s r s r s r s Let D n = group of symmetries of a regular n-gon, where symmetries means ways to move the n-gon so that the outline ends looking the same, but the vertices have moved. Some properties: () There are always n symmetries, i.e. D n has order n. () The symmetries are, for example, generated by r = rotate clockwise (0/n) and s = flip over a vertical axis. () The element r has order n, andtheelements has order. () The elements r and s don t commute, but they do satisfy rs = sr and sr = r s. Group presentations A subset of elements S G with the property that every element of G can be written as a finite product of elements of S and their inverses is called a set of generators of G. WewritehSi = G. Ex: D n is generated by S = {r, s}; Z is generated by ; Z/nZ is generated by. Any equations that are satisfied in G are called relations. Ex: The generators S = {r, s} satisfy s = r n =and rs = sr. If a set of relations R has the property that any relation in G can be derived from those in R then those generators and relations form a presentation of G, written hgenerators relationsi. In short, a presentation is everything you need to build the group. Some examples: D = hr, s r = e, s = e, r s = sri Z/Z = h = ei (Yes, weird notation!) Z = h ;i = hi (If there are no relations, i.e. R = ;, wewriteg = hsi.)

4 Intuition from linear algebra Generators are like spanning sets from linear algebra. For example, let G = Z.Thenx =(, 0) generates and also x + x =(, 0),x+ x + x =(, 0),..., x =(, 0) x + x =(, 0),... Throwing in y =(0, ) you also get y =(0, ),x+ y =(, ), etc.. So S = {x, y} generates Z. The only additional information you need to define the group is that xy = yx. So Z = hx, y xy = yxi. A minimum set of generators is like a basis from linear algebra. CAUTION!! Minimum versus minimal: Z = hi = h, i. Example Let G be the group Now a = a and b = b. G = ha, b a = b = e, aba = babi Other ways of writing aba = bab: abab = ba ababa = b ababab = e bbaaaba = bab Claim: G has elements.

5 Presentation problem Question: When are two presentations hs R i and hs R i actually presentations for the same group? HARD Claim: If we let a = s and b = rs, then ha, b a = b = e, aba = babi = hr, s r = s = e, rs = sr i = D

6 The symmetric group Let X be a finite non-empty set, and let S X be the set of bijections from the set to itself, i.e. the set of permutations of the elements. For example, if X = {,, } then S X contains S X forms a group under function composition. I A permutation followed by another permutation is, which is itself a permutation (binary operation) I Function composition is associative. I The bijection x! x for all x X serves as the identity. I Every bijection is invertible. The group S X is called the symmetric group on X. The symmetric group When X =[n] ={,,...,n} we denote S X by S n,andcallitthe symmetric group of degree n. Fact: It turns out that S X is essentially the same group as S X. (Later: they are isomorphic.) Proposition The order of S n is S n = n!.

7 Some notation Permutations can be represented in many ways: = means () =, () =, etc. (Cauchy s) two-line notation: = One-line notation: = Cycle notation: denoted by ()()() or just ()() Multiplication of diagrams: = =

8 Cycles A cycle is a string of integers what represents the element of S n that cyclically permutes these integers (and fixes all others). Specifically, a! a a! a (a a...a`) sends. a`! a Example: () is the permutation in S n that sends to, to, to, and everything else to itself. Every permutation can be expressed as the product (composition) of cycles, usually in several ways. Example: () = () = ()()() (draw the permutation diagrams) Algorithm for writing a permutation in cycles:. Start with (.. If a is the last element of the cycle, either: (i) If (a) is the first element of the cycle, close the cycle. If there are any numbers left unused, start a new cycle with the least available number. (ii) If (a) is the not first element of the cycle, add (a) next in the cycle.. Repeat until all numbers,...,n appear in some cycle.. Delete any cycles of length. = =

9 Multiplying cycles: Algorithm:. Start the first cycle of your answer with a =.. If the last number that you wrote in your answer is a, letx = a.. Look for the rightmost cycle with a x in it, and reset x to its successor in that cycle.. Always moving left from cycle-to-cycle, look for the next cycle with x in it and replace it with it s successor.. When you run out of cycles, write x as the successor of a in your answer, unless.... If x was the first element of the current cycle of your answer, then close the cycle, and start a new one with the least number not appearing yet in your answer. If = () and = ()() as before, then = ()()() = You try: (a) Write in cycle notation: = = (b) Draw the maps (like the diagrams above) for = ()() = ()()() (c) Use the cycle notation to compute and.checkusing the diagrams (stack on top of and resolve).