## Due June 5 2019

### Dihedral Groups

Recall that in lecture we saw that Dn, the dihedral group of order 2n, is generated by a fundamental rotation, r, and a reflection, s. We express this fact by writing Dn = (r, s). We saw that every element is equal to either rk (which we call a rotation) or rks (which we call a reflection) for some k with 0 k n 1.

1. List all of the subgroups of D6by providing generators for those subgroups (ie like (r)国外留学生代写
2. How many elements of order 2 are in Dn? (the answer will depend onn)
3. Prove that every subgroup of Dneither contains no reflections, or contains as many reflections as rotations.
4. In lecture we sketched a proof that Dn= e, r, r2, . . . , rn1, s, rs, r2s, . . . rn1s . Adapt that sketch into a proper proof by induction, using the definition of r, s , and the fact that r and s satisfy the relations rn = e, s2 = e, and  srs r1.

### Permutations and Symmetric Groups  国外留学生代写

1. Consider the group SZ=   f : Z Z f bijective . Which of the following are subgroups? For those that are, prove they are with a test, and for those that aren’t show why not:

(a)  {f   SZ| f (n> 0 if n > 0},

(b){f  SZ| f (n> 0 if an only if n > 0},

(c){f  SZ| f (n) = n if n is even}.

1. Suppose that α S6 is given in cyclic notation by: (123)(456), and β is given in array notation by

1    2    3    4    5  6

3    4    5    6    1  2

Calculate αβ and βα. Recall that as we view elements of Sn as functions on the set 1, 2, . . . , n , their composition is calculated from right to left.

1. What are the orders of the following elements S9, given in cyclic notation:

(a)(12)(3456)(78),

(b) (123)(4567),

(c) (456)(789)(123),

1. Write down an element of S9that has the highest possible order of an element in S9. Explain why you know it has the highest possible

### Conjugations and Centralizers  国外留学生代写

1. What is the centralizer of r in D5? What is the centralizer of s inD5?
2. What is the centralizer of (12)(34) in S6? You can give the answer by providing generators, or by writing out all of the elements in the
3. Prove  thatZ(tt) = aG C(a). (Recall Z(tt) is the center of tt and C(a) is the centralizer of a)
4. Ifα and τ are permutations in S5, and α = (12)(345), and τ = (135)(24), calculate τ ατ 1.
5. Let α S6be given in cyclic notation by (12)(34)(56). Find elements β and γ in S6 so that:
• αβ = βα, αγ = γα, and βγ =γβ,
• βƒ∈ (α, γ)γ ƒ∈ (α, β), and α ƒ∈ (β, γ).
1. Repeatthe previous question but instead the condition that γβ βγ is replaced by γβ βγ. What does this say about the centralizer of (12)(34)(56)?

### Alternating Group

1. Determinethe parity (even or odd) of the following permutations: (123)(456), (12)(34)(56), (1946)(783). Which of them are in A9?
2. If α and β are permutations in Sn, prove that if αβ is an odd permutation then either α is odd, or β is odd, but not both.
3. Prove that Sncannot be generated by 3-cycles.  国外留学生代写
4. Write (491836) S9as a product of transpositions. (Recall a transposition is a 2-cycle)
5. Write(37) S9 as a product of elementary transpositions. (Recall that an elementary transposition is one of the form (i, i + 1).
6. Explain how you could use the idea behind the previous two problems to write any element of Snas a product of elementary transpositions.

### Lagrange’s Theorem  国外留学生代写

1. Write the cosets of (r2) inD6.
2. Find all subgroups of D8whose index is equal to its order.
3. Supposep is a prime number and tt is a group with |tt| p.  Prove that tt is cyclic.
4. If tt has order pq, with p and q distinct prime numbers, prove that all proper subgroups of tt are cyclic. Give an example of such a tt that is not a cyclicgroup.