Math 3121 Abstract Algebra I Lecture 7: Finish Section 7 Sections 8.
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Transcript of Math 3121 Abstract Algebra I Lecture 7: Finish Section 7 Sections 8.
Math 3121Abstract Algebra I
Lecture 7:Finish Section 7
Sections 8
Finish Section 7
• Examples in class of Cayley Digraphs
Cayley Diagraph
• For each generating set of a finite group G, we can draw a graph whose vertices are elements of G and whose arcs represent right multiplication by a generator. Each arc is labeled according to the generator.
• Examples in class: Z6
Properties of Cayley Diagraphs1. Can get to any vertex from any other by a path. Reason: Every equation
g x = h has a solution in G and each member of G can be written as a product of generators and their inverses.
2. At most one arc goes from any vertex g to a vertex h. Reason: The solution of g x = h is unique.
3. Each vertex g has exactly one arc of each type starting at g and exactly one arc of each type ending at g. Reason: It is constructed this way.
4. If two different sequences of arc types go from vertex g to vertex h, then these two sequences applied to any other vertex will go to the same vertex.
Note: These four properties characterize Cayley diagraphs.
Examples
• Examples: Write out table of group described by Cayley
diagraph on page 72. Note the inner and outer squares have different directions.
Try this with triangles - note the directions of inner and outer triangles have same direction on page 71. Now do them with opposite directions.
What about pentagons?
HW for Section 7
• Don’t hand in:Pages 72-73: 1, 3, 5, 9
• Hand in (Due Tues, Oct 28):page 73: 12, 14, 16
Section 8
• Section 8: Groups of Permutations– Definition and Notation of Permutation– Theorem: Permutations on a set form a group
with composition as binary operation.– Definition: Symmetric Group on n letters– Definition: Dihedral group– Cayley’s Theorem
Permutations
Definition: A permutation of a set A is a function from A to A that is one-to-one and onto.
Examples:1) Let A = {a, b, c} , and let f: A A such that
f(a) = bf(b) = cf(c) = a
2) Let A = the set of real numbers, and let f: A A such that f(x) = 2 x. Does this work if A is the set of integers?
Permutation Groups
Theorem: Let A be a nonempty set, and let Perm[A] be the set of permutations of A. Then Perm[A] is a group with composition as the binary operation.
Proof: Composition is a well defined binary operation on Perm[A]. It satisfies: 1) It is associative because composition is associative.2) The identity map from A to itself acts as an identity for composition. Hence Perm[A] has an identity. 3) Every permutation is one-to-one onto, and thus has an inverse function. The inverse is also a 1-1 function from A onto itself. Hence Perm[A] is closed under inverses.
Permutation Notation
Write
for f: {x1, x2, …, xn} {x1, x2, …, xn}
Note: Most of the time, the set will be {1, 2, …,n}
)(...)()(
...
21
21
n
n
xfxfxf
xxx
Composition in Permutation Notation
))((...))(())((
...
)(...)()(
...
)(...)()(
...
21
21
21
21
21
21
n
n
n
n
n
n
xfgxfgxfg
xxx
xfxfxf
xxx
xgxgxg
xxx
• Composition in permutation notation is represented by multiplicatively. Note the order is right to left in this textbook (and hence in this class).
Composition in Permutation Notation
))((...))(())((
...
)(...)()(
...
)(...)()(
...
21
21
21
21
21
21
n
n
n
n
n
n
xfgxfgxfg
xxx
xfxfxf
xxx
xgxgxg
xxx
Procedure: Fill in each column at a time. For each column of the result, the top row determines a column of the right system. In that column, the entry in the second row determines a column of the left system. In that column, the entry in the second column determines the entry of the selected column of the result.
Example
• Try
4321
2143
4321
4132
4321
The Symmetric Group on n Letters
• Sn denotes the permutation group on the set {1, 2, …, n} and is called the symmetric group on n letters.
• Note: Sn has n! elements. Why?
• Look at S1, S2 , S3
S3 = Symmetric Group on 3 Letters
ρ0 ρ1 ρ2 μ1 μ2 μ3
ρ0 ρ0 ρ1 ρ2 μ1 μ2 μ3
ρ1 ρ1 ρ2 ρ0 μ3 μ1 μ2
ρ2 ρ2 ρ0 ρ1 μ2 μ3 μ1
μ1 μ1 μ2 μ3 ρ0 ρ1 ρ2
μ2 μ2 μ3 μ1 ρ2 ρ0 ρ1
μ3 μ3 μ1 μ2 ρ1 ρ2 ρ0
312
321
123
321
231
321
213
321
132
321
321
321
3
2
1
2
1
0
Symmetries of the Equilateral Triangle
ρ01
3
2
μ31
3
2μ2
1
3
2μ1
1
3
2
ρ11
3
2ρ2
1
3
2
S3=D3 = 3rd Dihedral Group
ρ0 ρ1 ρ2 μ1 μ2 μ3
ρ0 ρ0 ρ1 ρ2 μ1 μ2 μ3
ρ1 ρ1 ρ2 ρ0 μ3 μ1 μ2
ρ2 ρ2 ρ0 ρ1 μ2 μ3 μ1
μ1 μ1 μ2 μ3 ρ0 ρ1 ρ2
μ2 μ2 μ3 μ1 ρ2 ρ0 ρ1
μ3 μ3 μ1 μ2 ρ1 ρ2 ρ0
312
321
123
321
231
321
213
321
132
321
321
321
3
2
1
2
1
0
1
3
2
S4=D4 = Dihedral Group
ρ0 ρ1 ρ2 ρ3 μ1 μ2 δ1 δ2
ρ0 ρ0 ρ1 ρ2 ρ3 μ1 μ2 δ1 δ2
ρ1 ρ1 ρ2 ρ3 ρ0 δ1 δ2 μ2 μ1
ρ2 ρ2 ρ3 ρ0 ρ1 μ2 μ1 δ2 δ1
ρ3 ρ3 ρ0 ρ1 ρ2 δ2 δ1 μ1 μ2
μ1 μ1 δ2 μ2 δ1 ρ0 ρ2 ρ3 ρ1
μ2 μ2 δ1 μ1 δ2 ρ2 ρ0 ρ1 ρ3
δ1 δ1 μ1 δ2 μ2 ρ1 ρ3 ρ0 ρ2
δ2 δ2 μ2 δ1 μ1 ρ3 ρ1 ρ2 ρ0
2341
4321
4123
4321
1234
4321
3412
4321
3214
4321
2143
4321
1432
4321
4321
4321
2
1
2
1
3
2
1
0
1
3
2
4
Symmetries of the Square
3ρ04
21
3ρ14
21
3ρ24
21
3ρ34
21
3μ14
21
3μ24
21
3δ14
21
3δ24
21
Cayley’s TheoremTheorem (Cayley’s Theorem): Every group is isomorphic to a group of
permutations.Proof: Let G be a group. We show that G is isomorphic to a subgroup of the
permutations on the set G. For each a in G, let ρa be the map from G to itself given by left multiplication by a. That is, ρa(g) = a g. Then ρa is one-to-one and onto. In fact, it has an inverse. So ρa is a permutation of the set G. The map ρ that takes a to ρa is a homomorphism, because
ρa b(g) = a b g = a ρb(g) = ρa (ρb(g)) = (ρa ρb)(g) ρ is one-to-one and it is onto its image f[G]. It is straightforward to show that f[G] is a subgroup of Perm[G]. Thus f induces an isomorphism between G and f[G].
Right and Left Regular Representations
• ρx(g) = g x as in the theorem defined the left regular representation f of G.
• Multiplication on the right gives the property that is not homomorphism: μx y = μy μx. This is sometimes called the antihomomorphism property. The inverse map reverses the order. So
• μx(g) = g x-1 defines a map that has the correct order to be a homomorphism. This is called the right regular representation f of G.
Examples
• Find the left and right representations of Z2 , Z3 , Z4, S3
HW for Section 8
• Don’t hand in: pages 83-84: 1, 3, 5, 7, 9, 11, 13• Hand in (Due, Tues, Oct 28): Page 84: 18