国外留学生代写-MAT301代写-Problem Set代写

MAT301, Problem Set 2

Due June 5 2019

国外留学生代写 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.

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.

 

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