Lecture 3 Presentations and more Great Groups
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1 Lecture Presentations and more Great Groups
2 From last time: 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. We write S = G. Example: D n is generated by S = {r, s}.
3 From last time: 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. We write S = G. Example: D n is generated by S = {r, s}. Any equations that are satisfied in G are called relations. Example: The generators S = {r, s} satisfy s = r n = and rs = sr.
4 From last time: 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. We write S = G. Example: D n is generated by S = {r, s}. Any equations that are satisfied in G are called relations. Example: 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 generaors relations. In short, a presentation is everything you need to build the group.
5 Intuition from linear algebra Generators are like spanning sets from linear algebra.
6 Intuition from linear algebra Generators are like spanning sets from linear algebra. For example, let G = (Z ). Then x = (, 0) generates x + x = (, 0), x + x + x = (, 0),..., and also x = (, 0), x + x = (0, 0),...
7 Intuition from linear algebra Generators are like spanning sets from linear algebra. For example, let G = (Z ). Then x = (, 0) generates and also x + x = (, 0), x + x + x = (, 0),..., x = (, 0), x + x = (0, 0),... Throwing in y = (0, ) you also get So S = {x, y} generates Z. y = (0, ), x + y = (, ), etc..
8 Intuition from linear algebra Generators are like spanning sets from linear algebra. For example, let G = (Z ). Then x = (, 0) generates and also x + x = (, 0), x + x + x = (, 0),..., x = (, 0), x + x = (0, 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 = x, y xy = yx.
9 Intuition from linear algebra Generators are like spanning sets from linear algebra. For example, let G = (Z ). Then x = (, 0) generates and also x + x = (, 0), x + x + x = (, 0),..., x = (, 0), x + x = (0, 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 = x, y xy = yx. A minimum set of generators is like a basis from linear algebra. CAUTION!! Minimum versus minimal: Z = =,.
10 Example Let G be the group G = a, b a = b =, bab = a
11 Example Let G be the group Now a = a and b = b G = a, b a = b =, bab = a
12 Example Let G be the group G = a, b a = b =, bab = a Now a = a and b = b Other ways of writing bab = a: baba = abab = abb = ba bba = ab
13 Example Let G be the group G = a, b a = b =, bab = a Now a = a and b = b Other ways of writing bab = a: baba = abab = abb = ba bba = ab Also, aba = b aba = b baba = b
14 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.
15 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
16 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.
17 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. A permutation σ followed by another permutation τ is τ σ, which is itself a permutation (binary operation)
18 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. A permutation σ followed by another permutation τ is τ σ, which is itself a permutation (binary operation) Function composition is associative.
19 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. A permutation σ followed by another permutation τ is τ σ, which is itself a permutation (binary operation) Function composition is associative. The bijection x x for all x X serves as the identity.
20 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. A permutation σ followed by another permutation τ is τ σ, which is itself a permutation (binary operation) Function composition is associative. The bijection x x for all x X serves as the identity. Every bijection is invertible. The group S X is called the symmetric group on X.
21 The symmetric group When X = [n] = {,,..., n} we denote S X by S n, and call it the symmetric group of degree n.
22 The symmetric group When X = [n] = {,,..., n} we denote S X by S n, and call it the symmetric group of degree n. Fact: It turns out that S X is essentially the same group as S X.
23 The symmetric group When X = [n] = {,,..., n} we denote S X by S n, and call it the symmetric group of degree n. Fact: It turns out that S X is essentially the same group as S X. Proposition The order of S n is S n = n!.
24 Some notation Permutations can be represented in many ways: σ = means σ() =, σ() = 4, etc
25 Some notation Permutations can be represented in many ways: σ = means σ() =, σ() = 4, etc (Cauchy s) two-line notation: ( ) σ =
26 Some notation Permutations can be represented in many ways: σ = means σ() =, σ() = 4, etc (Cauchy s) two-line notation: ( ) σ = One-line notation: σ = 4765
27 Some notation Permutations can be represented in many ways: σ = means σ() =, σ() = 4, etc (Cauchy s) two-line notation: ( ) σ = One-line notation: σ = 4765 Cycle notation: (best for multiplication):
28 Some notation Permutations can be represented in many ways: σ = means σ() =, σ() = 4, etc (Cauchy s) two-line notation: ( ) σ = One-line notation: σ = 4765 Cycle notation: (best for multiplication): denoted by (4)(57)(6) or just (4)(57)
29 Try it: Write in cycle notation: σ = σ = Draw the maps (like the diagrams above) for τ = (7)(5) τ = ()(4)(56) Use the cycle notation to compute σ σ and τ τ. Check using the diagrams (stack σ on top of σ and resolve).
30 Definition The length of a cycle is the number of integers appearing in the cycle. Two cycles are disjoint if they have no numbers in common.
31 Definition The length of a cycle is the number of integers appearing in the cycle. Two cycles are disjoint if they have no numbers in common. Claim The decomposition of a permutation as the product of disjoint cycles is unique up to ordering of the cycles.
32 Definition The length of a cycle is the number of integers appearing in the cycle. Two cycles are disjoint if they have no numbers in common. Claim The decomposition of a permutation as the product of disjoint cycles is unique up to ordering of the cycles. Fact S n is non-abelian for n. However, disjoint cycles pairwise commute.
33 Definition The length of a cycle is the number of integers appearing in the cycle. Two cycles are disjoint if they have no numbers in common. Claim The decomposition of a permutation as the product of disjoint cycles is unique up to ordering of the cycles. Fact S n is non-abelian for n. However, disjoint cycles pairwise commute. Claim The order of a permutation is lowest common multiple of its cycle lengths.
34 A presentation for S n Let s i be the transposition s i = (i i + ) that switches i and i + and leaves everything else fixed. Then S = {s i i =,..., n } = {(), (), (4),..., (n n)} generates S n.
35 A presentation for S n Let s i be the transposition s i = (i i + ) that switches i and i + and leaves everything else fixed. Then S = {s i i =,..., n } = {(), (), (4),..., (n n)} generates S n. Some relations: s i = s i s j = s j s i if i j ± s i s i+ s i = s i+ s i s i+ for i =,..., n.
36 A presentation for S n Let s i be the transposition s i = (i i + ) that switches i and i + and leaves everything else fixed. Then S = {s i i =,..., n } = {(), (), (4),..., (n n)} generates S n. Some relations: s i = s i s j = s j s i if i j ± s i s i+ s i = s i+ s i s i+ for i =,..., n. Claim These generators and relations form a presentation for S n. (Maybe we ll prove this later)
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