## School of Mathematics and Statistics

## MT4003 Groups

## Problem Sheet V: Constructing Groups

数学math代写 Construct three non-abelian groups of order 24 that are pairwise non-isomorphic. Prove that they are indeed not isomorphic to each other.

**1.A permutation group ***G ≤* *S*_{X}* ***is called ***transitive ***if for all ***x, y ∈** **X ***there exists some ***σ ∈** **G ***such that ***xσ ***= ***y***.**

*G ≤*

*S*

_{X}

*transitive*

*x, y ∈*

*X*

*σ ∈*

*G*

*xσ*

*y*

**(a) Prove that every group is isomorphic to a transitive subgroup of some symmetric group. **

**(b) What is the smallest positive integer ***n ***such that a cyclic group ***C***6 ****of order 6 is isomorphic to ****a transitive subgroup of ***S*_{n}**? **

**(c) What is the smallest positive integer ***n ***such that a cyclic group ***C***6 ****of order 6 is isomorphic to ****a subgroup of ***S*_{n}**? **

**2.Let ***n ***be a positive integer. If ***σ ***is an arbitrary permutation in ***S_{n}*

**, defifine**

*∈*

*S*

_{n}

_{+2}**by**

**Show that the map Ø****: ***S_{n}*

*→*

*S*_{n}_{+2}

**by**

*σØ***=**

**is an injective homomorphism with im Ø**

*≤*

*A*_{n}_{+2}**.**

**Deduce that every fifinite group is isomorphic to a subgroup of an alternating group. **

**3.Let ***F ***be a fifield and let ***V ***= ***F*^{n}* ***be the vector space of row vectors with entries from ***F ***. Let 数学math代写**

*F*

*V*

*F*

^{n}

*F*

* ***= ***{***e****1***, ***e****2***,…, ***e***n**} ***be the standard basis for ***V ***. **

**(a) If ***σ ***is a permutation in ***S_{n}*

**, defifine a linear map**

*T*

_{σ}

**:**

*V →*

*V***by**

**e**_{i}*T_{σ}*

**= e**

_{iσ }**for**

*i***= 1, 2, . . . ,**

*n.***Show that ***T_{σ}*

*T*

_{τ}

**=**

*T*_{σ}

_{τ}**for all**

*σ, τ ∈*

*S*_{n}**.**

**(b) As we are writing maps on the right, the matrix of a linear transformation ***T ***with respect to **

**is obtained by expressing**

_{e}

_{i}

*T***in terms of the basis and writing the coeffiffifficients along the**

*rows***of the matrix.**

**Let ***A*_{σ}* ***denote the matrix of ***T*_{σ}* ***with respect to ****. Describe the entries of ***A*_{σ}**. (That is, specify ****which entries equal 1 and which equal 0.) **

**(c) Show that the map Ø****: ***S_{n}*

*→*

**GL**

*n***(**

*F***) by**

*σ*

*→*

*A*

*σ***is an injective homomorphism.**

**(d) Deduce that every fifinite group is isomorphic to a subgroup of GL***n***(***F ***) for some positive inte-****ger ***n***. **

**4.Let ***G ***and ***H ***be any groups. 数学math代写**

**(a) Show that ***G **× **H ≅*** ***H ×** **G***. **

**(b) Show that ***G ≅*** ***G ×** ***1. **

**(c) Show that ***G ×** **H ***is abelian if and only if both ***G ***and ***H ***are abelian. **

**(d) Show that if ***g ∈** **G ***and ***h ∈** **H***, then **

*o (***(***g, h***) )** **= lcm(***o***(***g***)***, o***(***h***))***. *

**[The left-hand side denotes the order of the element (***g, h***) in the direct product ***G ×** **H***.]**

**5.Construct three non-abelian groups of order 24 that are pairwise non-isomorphic. Prove that they ****are indeed not isomorphic to each other. 数学math代写**

6.**(a) Show that ***Q***8 ****is directly indecomposable (that is, it cannot be written as ***Q***8 ≅***M×**N ***for two ****non-trivial groups ***M ***and ***N***). **

**(b) Show that a cyclic group ***C**p**n ***of prime-power order is directly indecomposable. [Hint: How ****many subgroups of order ***p ***does it possess?] **

**7.Consider the direct product ***G ***= ***S***3 ×**

*S***3**

**of two copies of the symmetric group of degree 3.**

**(a) Let ***N ***= ***A***3 ×**

*A***3**

**. Find all normal subgroups of**

*G***that are contained in**

*N***. [Hint: First describe**

**all the subgroups of**

*N***. What is the conjugate of an element (**

*x, y***) by an element (1**

*, g***)?]**

**(b) How many normal subgroups of ***G ***are there containing ***N***? **

**(c) Find all the normal subgroups of ***G ***and hence all quotient groups of ***G***. **

**8.Let ***n ∈** *N **be odd, and consider the dihedral group ***D*_{4}_{n }**of order 4***n***, with the standard generators α ****and ***β***. Let ***H ***= 〈α**^{n}*〉** ***and ***K ***= 〈α****2***, β〉***. Prove that: 数学math代写**

*n ∈*

*D*

_{4}

_{n }*n*

*β*

*H*

^{n}*〉*

*K*

*, β〉*

**(a) both ***H ***and ***K ***are normal subgroups of ***D_{4}_{n}*

**;**

**(b) ***H ≅*** **Z_{2}**and ***K ≅*** ***D_{2}_{n}*

**;**

**(c) ***H ∩** **K ***= ***{***( )***} ***and ***HK ***= ***D_{4}_{n}*

**;**

**and conclude that ***D_{4}_{n}*

*≅***Z**

_{2}

*×*

*D*_{2}_{n}**.**

**9.Let ***G***1****, ***G***2****, …, ***G**k ***be groups and let ***D ***= ***G*_{1}** ,*** **G*_{2}** ×**

*··· ×*

*G*

_{k}

**be their direct product.**

**(a) Show that ***D ***is indeed a group. **

**(b) Find a homomorphism from ***G_{i}*

**with image**

_{i}**=**

*{***(1**

*,…,***1**

*, g,***1**

*,…,***1)**

*|*

*g ∈*

*G*_{i}

*}***. Show that**

_{i}**is a normal subgroup of ***D ***isomorphic to ***G_{i}*

**.**

**(c) Describe the quotient group ***D/ ***_{i}**. Justify your answer. 数学math代写

**(d) Show that ***D ***= **_{1}** **_{2}** ***… *_{k}**. **

**(e) Show that **

_{i}* ∩** ***( **_{1}*… *_{i-1 }_{i+1}*… ***_{k}**) = 1 for all

*i***.**

**10.Let ***G ***be a group possessing normal subgroups ***N_{1}*

**,**

*N*_{2}**, …,**

*N*

*k***such that**

*G***=**

*N*_{1}

*N*_{2}

*…N*_{k}

**and**

*N_{i}*

*∩*

**(**

*N*_{1}

*…N*_{i-1}

*N*_{i+1}

*…N*_{k}**) = 1 for all**

*i***. Show that**

*G ≅*** ***N*_{1}** ×*** **N*_{2}** ×*** ··· × **N**k**.*

**11.Show that the converse of Proposition 5.12 is not true. That is, fifind groups ***G***1 ****and ***G***2 ****and a normal ****subgroup ***P **G_{1}*

**×**

*G*_{2}

**such that**

**≠**

*P*

*M ×*

*N***for any**

*M*

*G*_{1}

**and**

*N*

*G*_{2}**.**

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