## School of Mathematics and Statistics

## MT4003 Groups

## Problem Sheet I: Defifinition and Examples of Groups

理科数学作业代写 Let F5 = {0, 1, 2, 3, 4} be the fifield of order 5, where addition and multiplication are calculated modulo 5. You may assume that F5 is a fifield.

**1.Consider the following two permutations from the symmetric group ***S***7 ****of degree 7: 理科数学作业代写**

*S*

**Write each of the following permutations as products of disjoint cycles: (i) α**

**; (ii)**

*β***; (iii)**

*α*

*β***;**

**(iv) ***βα***; (v) α**

^{−}

^{1}**; (vi)**

*β*

^{−}

^{1}**; (vii) (α**

*β***)**

^{−}^{1}**; (viii)**

*β*^{−}^{1}**α**

^{−}^{1}**.**

**2.Show that the symmetric group ***S**n ***is non-abelian if and only if **** n **>

**3. 理科数学作业代写**

**3.Let **F**5 ****= ***{***0***, ***1***, ***2***, ***3***, ***4***} ***be the fifield of order 5, where addition and multiplication are calculated ****modulo 5. You may assume that **F**5 ****is a fifield. **

**Consider the general linear group GL****2****(**F**5****), more often denoted by GL****2****(5). **

**(a) For ****, compute ***AB ***and ***A^{−}^{1}*

**.**

**(b) Prove that GL****2****(5) is non-abelian. **

**(c) Determine the order of GL****2****(5). **

**[Hint: A matrix is invertible if and only if its rows (or, equivalently, columns) are linearly ****independent. How do you choose a linearly independent pair of vectors from **F^{2}_{5}**?]**

**4.Let ***F ***be an arbitrary fifield. Show that the general linear group GL***n***(***F ***) is non-abelian if and only if ***n ≥* **2. 理科数学作业代写**

*F*

*n*

*F*

*n ≥*

**5.Verify that the Klein 4-group is indeed a group. **

**[Do not check each of the 64 cases for associativity. Instead, use careful thought to reduce how many ****checks you need to perform.] **

**6.Consider the following three matrices with entries from the complex numbers: **

**(a) Let G = {I,-I,A,-A, B,-B,C,-C}****, where ***I ***denotes the usual 2 ***× ***2 identity matrix. Cal-****culate a multiplication table for ***G ***and hence show that matrix multiplication defifines a binary ****operation on ***G***. **

**(b) Deduce that the quaternion group is indeed a group.**

**7.Let ***G ***be a group and ***a ***be an arbitrary (but fifixed) element of ***G***. Defifine a mapping τ**_{a}* ***: ***G →**G ***by ***xτ*_{a}* ***= ***a*^{−}^{1}*xa ***for each ***x ∈** **G***. (This mapping is called ***conjugation by **a***.) **

*G*

*a*

*G*

_{a}

*G →*

*G*

*xτ*

_{a}

*a*

^{−}^{1}

*xa*

*x ∈*

*G*

*conjugation by*

*a*

**(a) Specialise (for this part only) to the case ***G ***= ***S***3 ****and ***a ***= (1 2 3). Compute ***στ**a ***for each ***σ ∈** **S*_{3}**. **

**(b) Prove that τ_{a}**

**is always an isomorphism (in any group).**

**(c) If ***G ***is an arbitrary group, fifind an element ***a ∈** **G ***such that τ**

*a***is the identity mapping (that is,**

*xτ*_{a}

**=**

*x***for all**

*x ∈*

*G***).**

**(d) Prove that ***G ***is abelian if and only if ***all ***the mappings ***⌧**a ***are equal to the identity mapping. **

**8.Let ***G***, ***H ***and ***K ***be groups. 理科数学作业代写**

**(a) If Ø****: ***G →** **H ***is an isomorphism, show that its inverse Ø***−***1 ****: ***H →** **G ***is also an isomorphism. **

**(b) If Ø****: ***G →** **H ***and** **: ***H →** **K ***are isomorphisms, show that Ø*** ***: ***G →** **K ***is also an isomorphism. **

**(c) Show that ≅**** (being isomorphic) is an equivalence relation on the class of all groups.**

**9.Let ***G ***be a group and defifine a mapping ***φ***: ***G →** **G ***by ***xØ ***= ***x*^{−}^{1}** ****for each ***x ∈** **G***. **

**(a) Prove that Ø**^{2}** ****(= ØØ****) is the identity mapping. **

**(b) Prove that Ø*** ***is a bijection. **

**(c) Prove that Ø*** ***is an isomorphism if and only if ***G ***is abelian. **

**(d) Suppose that ***G ***is a fifinite group of even order. Prove that there exists ***a ∈** **G ***of order 2. **

**10.Let ***G ***be any group. If ***x ∈** **G***, prove that (***x*^{−}^{1}**)**^{−}^{1}** ****= ***x***. 理科数学作业代写**

*G*

*x ∈*

*G*

*x*

^{−}^{1}

^{−}^{1}

*x*

**11.Let ***G ***be a group and suppose that ***x*^{2}** ****= 1 for all ***x∈**G***. Show that ***G ***is abelian. **

**12.Let ***G ***be a group and let ***a, b ∈** **G***. Prove that if ***a*^{2}**= 1 and ***b*^{2}*a ***= ***ab*^{3}**, then ***b^{5}*

**= 1.**

**[Hint: ***b^{4}*

**=**

*b*^{2}

*b*^{2}

*aa***,**

*b*^{6}

**=**

*b*^{2}

*b*^{2}

*b*^{2}**.]**

**13.Let ***G ***be a group. **

**(a) Prove that for any ***x, y ∈**G***, the following formulae involving the order of an element hold: 理科数学作业代写**

*o***(***x***) = ***o***(***x*^{−}^{1}**); ***o***(***x***) = ***o*

**(***y*^{−}^{1}*xy***), and ***o***(***xy***) = ***o***(***yx***). **

**(b) If ***σ ***is a permutation in ***S*_{n}**decomposed as a product of disjoint cycles as ***σ ***= ***σ*_{1}*σ*_{2}** ***… σ**k***, ****where ***σ**i ***is a cycle of length ***r**i***, then prove that **

*o***(***σ***) = lcm(***r***1***, r***2***,…,r**k***)***. *

**(c) Consider the matrices **

**from the general linear group GL****2****(**Q**). Show that ***o***(***A***) = 4, ***o***(***B***) = 3, but that ***AB ***has infifinite ****order. **

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