## Problem Sheet III: Normal Subgroups, Quotient Groups & Homomorphisms

### 1.Let G be a group and N be a subgroup of G. Show that the following conditions on N are equivalent:

(a) N is a normal subgroup of G.

(b) g1Ng ⊆ N for all g ∈ G.

(c) g1Ng = N for all g ∈ G.

(d) Ng = gN for all g ∈ G.

(e) Every right coset of N in G is also a left coset of N.

2.Let G be a group and N be a subgroup of index 2. Show that N is a normal subgroup of G. Is it true that a subgroup of index 3 is necessarily normal?

### 3.Let G be a group and suppose M and N are normal subgroups of G. Show that M \ N is a normal subgroup of G and that the set of products  新西兰数学代写

MN = { xy | x 2 M, y 2 N }

is a normal subgroup of G.

4.Let σ be any permutation in Sn and let

τ = (i1 i2 … ir)

be an r-cycle. Show that σ1τ σ is the r-cycle

(i1σ i2σ … irσ).

If an arbitrary permutation τ is written as a product of disjoint cycles, what does this tell you about the conjugate σ1τ σ?

### 5.Let G be a group, N be a normal subgroup of G and x be an element of G of fifinite order. Show that the element Nx of the quotient group G/N has fifinite order and that o(Nx) divides o(x).

6.Let N be the subgroup of S4 generated by (1 2)(3 4) and (1 3)(2 4).

(a) Find the elements of N and prove that N is normal in S4.

(b) List the elements of the quotient S4/N. Is S4/N cyclic? Is it abelian?  新西兰数学代写

### 7.Let G = S3 and H = h(1 2)i.

(a) Show that H is not a normal subgroup of S3.

(b) Show that H(1 3) = H(1 2 3) and H(2 3) = H(1 3 2).

(c) Is H(1 3)(1 3 2) = H(1 2 3)(2 3)?

(d) What do (b) and (c) illustrate?

8.Consider the quaternion group Q8.

(a) Find all orders of elements in Q8.

(b) Find all subgroups of Q8.

(c) Show that every subgroup of Q8 is normal.

(d) Describe, up to isomorphism, all homomorphic images of Q8.

### 9.Show that the alternating group A4does not possess a subgroup of order 6.  新西兰数学代写

What does this tell you about Lagrange’s Theorem?

10.(a) Give an example of a group G possessing two subgroups H and K with K ≤ H ≤ G, KH and H

G, but KG.

(b) Give an example of a group G with two subgroups H and K such that the set of products

HK = { hk | h 2 H, k 2 K }

is not a subgroup of G.

(c) Give an example of a group G with two normal subgroups K and L such that G/K ≅ G/L but

K L.

[Hint: All these happen fairly often and you know some fairly small examples of groups where it does happen!]  新西兰数学代写

### 11.Let G be a group and suppose that G has a chain of subgroups

G = G0 > G1 > ··· > Gn = 1

with the property that each Gi+1 is a normal subgroup of Gi and the quotient group Gi/Gi+1 is abelian for all i.

If H is any subgroup of G, show that H also has such a chain of subgroups with abelian quotients.

[Hint: Consider the groups H \ Gi and use the Second Isomorphism Theorem.]

12.We say that a group H is a homomorphic image of a group G if there exists a surjective homomor-phism G → H. Prove that every homomorphic image of an abelian group is abelian.