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.
- List all of the subgroups of D6by providing generators for those subgroups (ie like (r)) 国外留学生代写
- How many elements of order 2 are in Dn? (the answer will depend onn)
- Prove that every subgroup of Dneither contains no reflections, or contains as many reflections as rotations.
- In lecture we sketched a proof that Dn= e, r, r2, . . . , rn−1, s, rs, r2s, . . . rn−1s . 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 = r−1.
Permutations and Symmetric Groups 国外留学生代写
- 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}.
- 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.
- What are the orders of the following elements S9, given in cyclic notation:
(a)(12)(3456)(78),
(b) (123)(4567),
(c) (456)(789)(123),
- 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
- What is the centralizer of r in D5? What is the centralizer of s inD5?
- 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
- Prove thatZ(tt) = a∈G C(a). (Recall Z(tt) is the center of tt and C(a) is the centralizer of a)
- Ifα and τ are permutations in S5, and α = (12)(345), and τ = (135)(24), calculate τ ατ −1.
- 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).
- 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
- Determinethe parity (even or odd) of the following permutations: (123)(456), (12)(34)(56), (1946)(783). Which of them are in A9?
- 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.
- Prove that Sncannot be generated by 3-cycles. 国外留学生代写
- Write (491836) ∈ S9as a product of transpositions. (Recall a transposition is a 2-cycle)
- Write(37) S9 as a product of elementary transpositions. (Recall that an elementary transposition is one of the form (i, i + 1).
- 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
- Write the cosets of (r2) inD6.
- Find all subgroups of D8whose index is equal to its order.
- Supposep is a prime number and tt is a group with |tt| = p. Prove that tt is cyclic.
- 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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