Lecture 3
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 hSi = G.
Example: D2n 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 hSi = G.
Example: D2n is generated by S = {r, s}.
Any equations that are satisfied in G are called relations.
Example: The generators S = {r, s} satisfy s2 = rn = 1 and rs = sr−1 .
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 hSi = G.
Example: D2n is generated by S = {r, s}.
Any equations that are satisfied in G are called relations.
Example: The generators S = {r, s} satisfy s2 = rn = 1 and rs = sr−1 .
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
hgeneraors | relationsi.
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 = (Z2 ). Then x = (1, 0) generates
x + x = (2, 0), x + x + x = (3, 0), . . . ,
and also
x−1 = (−1, 0), x−1 + x = (0, 0), . . .
Intuition from linear algebra
Generators are like “spanning sets” from linear algebra.
For example, let G = (Z2 ). Then x = (1, 0) generates
x + x = (2, 0), x + x + x = (3, 0), . . . ,
and also
x−1 = (−1, 0), x−1 + x = (0, 0), . . .
Throwing in y = (0, 1) you also get
y −1 = (0, −1), x + y = (1, 1), etc..
So S = {x, y} generates Z2 .
Intuition from linear algebra
Generators are like “spanning sets” from linear algebra.
For example, let G = (Z2 ). Then x = (1, 0) generates
x + x = (2, 0), x + x + x = (3, 0), . . . ,
and also
x−1 = (−1, 0), x−1 + x = (0, 0), . . .
Throwing in y = (0, 1) you also get
y −1 = (0, −1), x + y = (1, 1), etc..
So S = {x, y} generates Z2 . The only additional information you
need to define the group is that xy = yx. So
Z2 = hx, y | xy = yxi.
Intuition from linear algebra
Generators are like “spanning sets” from linear algebra.
For example, let G = (Z2 ). Then x = (1, 0) generates
x + x = (2, 0), x + x + x = (3, 0), . . . ,
and also
x−1 = (−1, 0), x−1 + x = (0, 0), . . .
Throwing in y = (0, 1) you also get
y −1 = (0, −1), x + y = (1, 1), etc..
So S = {x, y} generates Z2 . The only additional information you
need to define the group is that xy = yx. So
Z2 = hx, y | xy = yxi.
A minimum set of generators is like a basis from linear algebra.
CAUTION!! Minimum versus minimal: Z = h1i = h2, 3i.
Example
Let G be the group
G = ha, b | a2 = b3 = 1, bab = ai
Example
Let G be the group
G = ha, b | a2 = b3 = 1, bab = ai
Now a−1 = a and b−1 = b2
Example
Let G be the group
G = ha, b | a2 = b3 = 1, bab = ai
Now a−1 = a and b−1 = b2
Other ways of writing bab = a:
baba = 1 abab = 1 abb = ba
bba = ab
Example
Let G be the group
G = ha, b | a2 = b3 = 1, bab = ai
Now a−1 = a and b−1 = b2
Other ways of writing bab = a:
baba = 1 abab = 1 abb = ba
Also, aba = b3 aba = b2 baba = b2
bba = ab
The symmetric group
Let X be a finite non-empty set, and let SX 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 SX be the set of bijections
from the set to itself, i.e. the set of permutations of the elements.
For example, if X = {1, 2, 3} then SX contains
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
The symmetric group
Let X be a finite non-empty set, and let SX be the set of bijections
from the set to itself, i.e. the set of permutations of the elements.
For example, if X = {1, 2, 3} then SX contains
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
SX forms a group under function composition.
The symmetric group
Let X be a finite non-empty set, and let SX be the set of bijections
from the set to itself, i.e. the set of permutations of the elements.
For example, if X = {1, 2, 3} then SX contains
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
SX forms a group under function composition.
I
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 SX be the set of bijections
from the set to itself, i.e. the set of permutations of the elements.
For example, if X = {1, 2, 3} then SX contains
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
SX forms a group under function composition.
I
I
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 SX be the set of bijections
from the set to itself, i.e. the set of permutations of the elements.
For example, if X = {1, 2, 3} then SX contains
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
SX forms a group under function composition.
I
I
I
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 7→ x for all x ∈ X serves as the identity.
The symmetric group
Let X be a finite non-empty set, and let SX be the set of bijections
from the set to itself, i.e. the set of permutations of the elements.
For example, if X = {1, 2, 3} then SX contains
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
SX forms a group under function composition.
I
I
I
I
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 7→ x for all x ∈ X serves as the identity.
Every bijection is invertible.
The group SX is called the symmetric group on X.
The symmetric group
When X = [n] = {1, 2, . . . , n} we denote SX by Sn , and call it the
symmetric group of degree n.
The symmetric group
When X = [n] = {1, 2, . . . , n} we denote SX by Sn , and call it the
symmetric group of degree n.
Fact: It turns out that SX is essentially the same group as S|X| .
The symmetric group
When X = [n] = {1, 2, . . . , n} we denote SX by Sn , and call it the
symmetric group of degree n.
Fact: It turns out that SX is essentially the same group as S|X| .
Proposition
The order of Sn is |Sn | = n!.
Some notation
Permutations can be represented in many ways:
1
2
3
4
5
6
7
σ=
means σ(1) = 3, σ(2) = 4, etc.
1
2
3
4
5
6
7
Some notation
Permutations can be represented in many ways:
1
2
3
4
5
6
7
σ=
means σ(1) = 3, σ(2) = 4, etc.
1
2
3
4
5
6
7
(Cauchy’s) two-line notation:
1 2 3 4 5 6 7
σ=
3 4 2 1 7 6 5
Some notation
Permutations can be represented in many ways:
1
2
3
4
5
6
7
σ=
means σ(1) = 3, σ(2) = 4, etc.
1
2
3
4
5
6
7
(Cauchy’s) two-line notation:
1 2 3 4 5 6 7
σ=
3 4 2 1 7 6 5
One-line notation: σ = 3421765
Some notation
Permutations can be represented in many ways:
2
1
3
4
5
6
7
σ=
means σ(1) = 3, σ(2) = 4, etc.
1
2
3
4
5
6
7
(Cauchy’s) two-line notation:
1 2 3 4 5 6 7
σ=
3 4 2 1 7 6 5
One-line notation: σ = 3421765
Cycle notation: (best for multiplication):
3
2
7
6
1
4
5
Some notation
Permutations can be represented in many ways:
2
1
3
4
5
6
7
σ=
means σ(1) = 3, σ(2) = 4, etc.
1
2
3
4
5
6
7
(Cauchy’s) two-line notation:
1 2 3 4 5 6 7
σ=
3 4 2 1 7 6 5
One-line notation: σ = 3421765
Cycle notation: (best for multiplication):
3
2
7
6
1
4
5
denoted by (1324)(57)(6)
or just (1324)(57)
Try it:
Write in cycle notation:
1
2
3
4
5
6
7
σ1 =
1
2
3
4
5
6
7
1
2
3
4
5
6
7
σ2 =
1
2
3
4
5
6
7
Draw the maps (like the diagrams above) for
τ1 = (137)(52)
τ2 = (12)(34)(56)
Use the cycle notation to compute σ1 σ2 and τ2 τ1 . Check using the
diagrams (stack σ2 on top of σ1 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
Sn is non-abelian for n ≥ 3. 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
Sn is non-abelian for n ≥ 3. However, disjoint cycles pairwise
commute.
Claim
The order of a permutation is lowest common multiple of its cycle
lengths.
A presentation for Sn
Let si be the transposition si = (i i + 1) that switches i and i + 1
and leaves everything else fixed.
Then
S = {si | i = 1, . . . , n − 1} = {(12), (23), (34), . . . , (n − 1 n)}
generates Sn .
A presentation for Sn
Let si be the transposition si = (i i + 1) that switches i and i + 1
and leaves everything else fixed.
Then
S = {si | i = 1, . . . , n − 1} = {(12), (23), (34), . . . , (n − 1 n)}
generates Sn .
Some relations:
s2i = 1
si sj = sj si
si si+1 si = si+1 si si+1
if i 6= j ± 1
for i = 1, . . . , n − 2.
A presentation for Sn
Let si be the transposition si = (i i + 1) that switches i and i + 1
and leaves everything else fixed.
Then
S = {si | i = 1, . . . , n − 1} = {(12), (23), (34), . . . , (n − 1 n)}
generates Sn .
Some relations:
s2i = 1
si sj = sj si
si si+1 si = si+1 si si+1
if i 6= j ± 1
for i = 1, . . . , n − 2.
Claim
These generators and relations form a presentation for Sn .
(Maybe we’ll prove this later)