Lecture Presentations and more Great Groups
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}.
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.
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.
Intuition from linear algebra Generators are like spanning sets from linear algebra.
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),...
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..
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.
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 = =,.
Example Let G be the group G = a, b a = b =, bab = a
Example Let G be the group Now a = a and b = b G = a, b a = b =, bab = a
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
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
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.
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
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.
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)
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 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.
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.
The symmetric group When X = [n] = {,,..., n} we denote S X by S n, and call it the symmetric group of degree n.
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.
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!.
Some notation Permutations can be represented in many ways: 4 5 6 7 σ = means σ() =, σ() = 4, etc. 4 5 6 7
Some notation Permutations can be represented in many ways: 4 5 6 7 σ = means σ() =, σ() = 4, etc. 4 5 6 7 (Cauchy s) two-line notation: ( ) 4 5 6 7 σ = 4 7 6 5
Some notation Permutations can be represented in many ways: 4 5 6 7 σ = means σ() =, σ() = 4, etc. 4 5 6 7 (Cauchy s) two-line notation: ( ) 4 5 6 7 σ = 4 7 6 5 One-line notation: σ = 4765
Some notation Permutations can be represented in many ways: 4 5 6 7 σ = means σ() =, σ() = 4, etc. 4 5 6 7 (Cauchy s) two-line notation: ( ) 4 5 6 7 σ = 4 7 6 5 One-line notation: σ = 4765 Cycle notation: (best for multiplication): 7 4 5 6
Some notation Permutations can be represented in many ways: 4 5 6 7 σ = means σ() =, σ() = 4, etc. 4 5 6 7 (Cauchy s) two-line notation: ( ) 4 5 6 7 σ = 4 7 6 5 One-line notation: σ = 4765 Cycle notation: (best for multiplication): 7 4 5 6 denoted by (4)(57)(6) or just (4)(57)
Try it: Write in cycle notation: 4 5 6 7 4 5 6 7 σ = σ = 4 5 6 7 4 5 6 7 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).
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.
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.
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.
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.
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.
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.
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)