Math 3121 Abstract Algebra I

1. Math 3121Abstract Algebra I Lecture 7: Finish Section 7 Sections 8

2. Finish Section 7 • Examples in class of Cayley Digraphs

3. 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

4. Properties of Cayley Diagraphs • 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. • At most one arc goes from any vertex g to a vertex h. Reason: The solution of g x = h is unique. • 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. • 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.

5. 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?

6. 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

7. 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

8. Permutations Definition: A permutation of a set A is a function from A to A that is one-to-one and onto. Examples: • Let A = {a, b, c} , and let f: A  A such that f(a) = b f(b) = c f(c) = a • 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?

9. 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.

10. Permutation Notation Write for f: {x1, x2, …, xn}  {x1, x2, …, xn} Note: Most of the time, the set will be {1, 2, …,n}

11. Composition in Permutation Notation • Composition in permutation notation is represented by multiplicatively. Note the order is right to left in this textbook (and hence in this class).

12. Composition in Permutation Notation 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.

13. Example • Try

14. 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

15. S3 = Symmetric Group on 3 Letters

16. Symmetries of the Equilateral Triangle 3 3 3 3 3 3 μ2 ρ1 μ1 ρ2 ρ0 μ3 2 2 2 2 2 2 1 1 1 1 1 1

17. S3=D3 = 3rd Dihedral Group 3 2 1

18. S4=D4 = Dihedral Group 4 3 1 2

19. Symmetries of the Square 4 3 4 3 4 3 4 3 ρ0 ρ2 ρ3 ρ1 1 2 1 2 1 2 1 2 4 3 4 3 4 3 4 3 δ1 δ2 μ2 μ1 1 2 1 2 1 2 1 2

20. Cayley’s Theorem Theorem (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].

21. 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.

22. Examples • Find the left and right representations of Z2 , Z3 , Z4, S3

23. 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