数学群论代考 MAT301 Week 7 Exercises

MAT301 Week 7 Exercises

数学群论代考 The universal property of π is useful for defining homomorphisms out of G/N. For each homomorphism φ : G → H with N ≤ ker φ

Recall that if G is a group we write H E G to express that “H is a normal subgroup of G”.

1 Examples of normal subgroups  数学群论代考

Exercise 1. Determine the normal subgroups of S3.

Exercise 2. Prove that the subgroup hri of rotations in Dn is a normal subgroup of dn.

Exercise 3. Prove that H = {id,(1 2)(3 4),(1 3)(2 4),(1 4)(2 3)} is a normal subgroup of A4. (It can be proved that H is the only proper and non-trivial subgroup of A4.)

Exercise 4 (A non-abelian group for which all subgroups are normal). Prove that every subgroup of Q8 is a normal.  数学群论代考

Exercise 5 (Subgroups of index 2 are normal). Let G be a group and let H be a subgroup of G of index 2 in G. Using the fact that there is only one left coset of H in G that is not equal to H and that there is only one right coset of H in G that is not equal to H, prove that H E G.

Exercise 6. Let G be a group and let H E G with |H| = 2. Prove that H ⊆ Z(G). (This exercise is included here because the preceding exercise might give you a hint for its solution and because it will help you with the next exercise.)

Exercise 7. Determine all normal subgroups of Dn of order 2.

Exercise 8 (The outer automorphism group). Let G be a group. For each g ∈ G, let Int(g) ∈ Aut(G) denote conjugation by g, i.e. for all x ∈ G we have Int(g)(x) = gxg−1 . Let Int(G) = {Int(g) : g ∈ G}, which we recall is called the inner automrophism group of G and its elements are called inner automorphisms of G. Prove that Int(G) E Aut(G). (The quotient group Out(G) = Aut(G)/ Int(G) is called the outer automorphism group of G even though its elements are not actually automorphisms of G.)

Exercise 9 (The centraliser of a normal subgroup). Let G be a group and let H E G.  数学群论代考
1. Prove that CG(H) E G.
2. Prove that G/CG(H) is isomorphic to a subgroup of Aut(H).

Exercise 10 (A5 is simple). Assuming that every non-trivial normal subgroup of A5 contains a 3-cycle (which you will prove in Assignment 4), prove that there does not exist a proper and non-trivial normal subgroup of A5. (A non-trivial group that does not have a proper and non-trivial normal subgroup is called a simple group. Therefore A5 is simple. It can be proved that for all n ≥ 5, every non-trivial normal subgroup of An contains a 3-cycle, and consequently that An is simple.)

2 Basic properties and constructions  数学群论代考

Exercise 11. Let G be a group and let H E G. Using the fact that G → G/H, g 7→ gH is a surjectivehomomorphism, prove that a quotient group of an abelian (resp. cyclic) group is abelian (resp. cyclic).

Exercise 12. Let G be a group.
1. Prove that if K ≤ H ≤ G and K E G, then K E H.
2. Prove that if H, K ≤ G and K E G, then K ∩ H E H

2.1 Intersections

Exercise 13 (Arbitrary intersections of normal subgroups are normal). Let G be a group and let {Hi}i∈I be an arbitrary set of normal subgroups of G. Prove that Ti∈I Hi is a normal subgroup of G.

Exercise 14 (The normal subgroup generated by a set). Let G be a group and let S be a subset of G. Define the normal subgroup of G generated by S (or normal closure of S in G) to be the subgroup

数学群论代考


  1. Prove that nclG(S) is the the unique minimal normal subgroup of G that contains S, that is, nclG(S) is the unique subgroup of G with the following properties: (i) nclG(S) is a normal subgroup of G containing S; (ii) if N is a normal subgroup of G containing S, then nclG(S) ⊆ N. For each subset T of G, define GT to be the set of conjugates of T in G, that is, GT = {gtg−1 : g ∈ G}.


  2. Prove that nclG(S) = h GSi, the subgroup of G generated by GS.  数学群论代考


2.2 Orders  数学群论代考

Exercise 15. Let G be a group, let H E G, and let a ∈ G.
1. Complete the following sentence. The order of aH in G/H is infinite if and only if a n ... for all positive integers n.
2. Suppose that aH has finite order in G/H. Complete the following sentence. The order of aH in G/H is the smallest positive integer n such that a n ... .  数学群论代考

Exercise 16. Let G be an abelian group and let H be the subgroup of G consisting of all elements of G of finite order. Prove that every non-identity element of G/H has finite order.
Exercise 17. Let G be a group, let H be a finite normal subgroup of G, and let n ∈ Z>0. Prove that if G/H has an element of order n, then G has an element of order n. (Note that if G = Z, H = 2Z, then G/H = Z/2Z has an element of order 2, but G = Z does not. Therefore the assumption that H is finite cannot be dropped.)

2.3 Subgroups of subgroups and characteristic subgroups

Exercise 18 (Normality is not transitive, i.e. K E H E G =6⇒ K E G)). Give an example of a group G, a normal subgroup H of G, and a normal subgroup K of G such that K is not a normal subgroup of G.

Exercise 19 (Characteristic subgroups). Let G be a group. A subgroup H ≤ G is said to be a characteristic subgroup of G if for all φ ∈ Aut(H) we have φ(H) = H. We write H char G to express that H is a characteristic subgroup of G.  数学群论代考


  1. Prove that H = {id,(1 2)(3 4),(1 3)(2 4),(1 4)(2 3)} is a characteristic subgroup of A4.

  2. Prove that every characteristic subgroup of G is a normal subgroup of G.

  3. Prove that a subgroup H of G is a characteristic subgroup of G if and only if for all φ ∈ Aut(G) we have φ(H) ≤ H.

  4. Prove that Z(G) is a characteristic subgroup of G.

  5. Prove that if K char H char G, then K char G, i.e. being a characteristic subgroup is a transitive relation

  6. Prove that if K char H E G, then K E G.

Exercise 20. Let G be a group and let N be a cyclic normal subgroup of G. Prove that if H ≤ N, then H E G.  数学群论代考

Exercise 21 (The derived or commutator subgroup). Let G be a group. Define G0 to be the subgroup of G generated by the set of commutators of elements of G. That is,

数学群论代考

The subgroup G0 is called the derived subgroup (or commutator subgroup) of G.
1. Prove that G0 is a characteristic subgroup of G, and hence a normal subgroup of G.
2. Prove that the group G/G0 is abelian.
3. If N E G and G/N is abelian, prove that G0 ≤ N.
4. Prove that if H ≤ G and G0 ≤ H, then H E G

2.4 Normalisers  数学群论代考

Exercise 22 (The normaliser of a set). Let G be a group and let S ⊆ G. We define the normaliser of S in G to be NG(S) = {g ∈ G : gSg−1 = S}.
1. Prove that CG(S) ≤ NG(S) ≤ G for all S ⊆ G.
2. Let H ≤ G. Prove that H is a normal subgroup of G if and only if NG(H) = G. (We say that H is a central subgroup of G if H ⊆ Z(G). We have a similar result: H is a central subgroup of G if and only if CG(H) = G.)
3. Let H ≤ G. Prove that H E NG(H).
4. Let S ⊆ G. Prove that CG(S) E NG(S).  数学群论代考
5. Let H ≤ G. Prove that NG(H)/CG(H) is isomorphic to a subgroup of Aut(H).

Exercise 23 (The product of two subgroups). Let G be a group and let H, K ≤ G. Recall that for subsets S, T ⊆ G, we define ST = {st : s ∈ S, t ∈ T}.
1. Prove that HK is a subgroup of G if and only if HK = KH, in which case it is the unique minimal subgroup of G that contains both H and K.
2. Prove that if H ⊆ NG(K) or K ⊆ NG(H), then HK = KH and is a subgroup of G.
3. Prove that if H and K are both normal subgroups of G, then HK is a normal subgroup of G.

2.5 The universal property of quotients

Exercise 24. Let G be a group, let N be a normal subgroup of G, and let π : G → G/N be the quotient homomoprhism, which is defined by π(g) = gN for all g ∈ G. The universal property of π is the following: for every group H and every homomorphism φ : G → H such that N ≤ ker φ, there exists a unique homomorphism φ˜ : G/N → H such that φ = φ˜ ◦ π.

The universal property of π is useful for defining homomorphisms out of G/N. For each homomorphism φ : G → H with N ≤ ker φ, we obtain a map φ˜ : G/N → H that we immdiately know is a well-defined homomorphism. Sometimes we will have a formula that we would like to use to define a homomorphism ψ : G/N → H, but we do not know a priori if ψ a well-defined homomorphism. We can conclude that it is, using the universal property of π, if we can show that ψ = φ˜ for some homomorphism φ : G → H such that N ≤ ker φ. In this exercise, we will prove the universal property of π. Let H be a group and let φ : G → H be a homomorphism with N ≤ ker φ.  数学群论代考


  1. Suppose that there exists a homomorphism φ˜ : G/N → H such that φ = φ˜ ◦ π. Prove that for all gN ∈ G/N we have φ˜(gN) = φ(g). Conclude that φ˜ is unique if it exists.

  2. Prove that the map φ˜ : G/N → H defined by φ˜(gN) = φ(g) for all gN ∈ G/N is a well-definedhomomorphism such that φ = φ˜ ◦ π. Conclude that the universal property of π holds.

世纪学院代写

更多代写:cs代写    计量经济代考   机器学习代写      r语言代写论文摘要代写

发表评论

客服一号:点击这里给我发消息
客服二号:点击这里给我发消息
微信客服1:essay-kathrine
微信客服2:essay-gloria