## School of Mathematics and Statistics

## MT4003 Groups

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

新西兰数学代写 If an arbitrary permutation τ is written as a product of disjoint cycles, what does this tell you about the conjugate σ−1τ σ?

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

*G*

*N*

*G*

*N*

**(a) ***N ***is a normal subgroup of ***G***. **

**(b) ***g*^{−}^{1}*Ng ⊆** **N ***for all ***g ∈** **G***. **

**(c) ***g^{−}^{1}*

*Ng***=**

*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**

*G*

*M*

*N*

*G*

*M*

*\*

*N*

*G*

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

**is a normal subgroup of ***G***. **

**4.Let ***σ ***be any permutation in ***S*_{n}* ***and let **

*τ ***= (***i*_{1}** ***i_{2}*

*… i*_{r}**)**

**be an ***r***-cycle. Show that ***σ**−***1 τ**

*σ***is the**

*r***-cycle**

**(***i***1***σ i***2***σ … i**r**σ***)***. *

**If an arbitrary permutation τ**

**is written as a product of disjoint cycles, what does this tell you about**

**the conjugate**

*σ*

^{−}

^{1}

*τ*

*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***). **

*G*

*N*

*G*

*x*

*G*

*Nx*

*G/N*

*o*

*Nx*

*o*

*x*

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

**(a) Find the elements of ***N ***and prove that ***N ***is normal in ***S***4****. **

**(b) List the elements of the quotient ***S*_{4}*/N***. Is ***S*_{4}*/N ***cyclic? Is it abelian? 新西兰数学代写**

**7.Let ***G ***= ***S***3 ****and ***H ***= ***h***(1 2)***i***. **

*G*

*S*

*H*

*h*

*i*

**(a) Show that ***H ***is not a normal subgroup of ***S*_{3}**. **

**(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 ***Q***8****. **

**(a) Find all orders of elements in ***Q***8****. **

**(b) Find all subgroups of ***Q***8****. **

**(c) Show that every subgroup of ***Q***8 ****is normal. **

**(d) Describe, up to isomorphism, all homomorphic images of ***Q***8****. **

### 9.**Show that the alternating group ***A*_{4}** ****does not possess a subgroup of order 6. 新西兰数学代写**

*A*

_{4}

**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***, ***K**H ***and ***H*

*G***, but ***K**G***. **

**(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*

*G*

*G ***= ***G***0 ***> G***1 ***> **··· **> G**n ***= 1 **

**with the property that each ***G**i***+1 ****is a normal subgroup of ***G**i ***and the quotient group ***G**i**/G**i***+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 **\ **G**i ***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. **

更多代写：统计网课代修推荐 线上考试怎么防止作弊 利物浦网课代上 Essay开头怎么写 Essays论文代写 Mathematics作业代写