**INTRODUCTION TO GROUP THEORY **

## MATH10010

## (Paper code MATH–10010)

数学代写价格 On this examination, the marking scheme is indicative and is intended only as a guide to the relative weighting of the questions.

**This paper contains two sections: Section A and Section B. **

**Each section should be answered in a separate booklet. **

Section A contains FIVE questions and Section B contains TWO questions

All SEVEN answers will be used for assessment.

Calculators are not permitted.

**Candidates may bring four sheets of A4 notes written double-sided into the ****examination. **

**Candidates must insert these into their answer booklet(s) for collection at the end ****of the examination. **

On this examination, the marking scheme is indicative and is intended only as a guide to the relative weighting of the questions.

**Section A: Short Questions 数学代写价格**

A1. (a) (**6 marks**) For the set and binary operation in each of (i) and (ii), state which of the axioms of a group are and are not satisfified. No explanations are necessary.

(i) The set of rational numbers Q with the binary operation of multiplication.

(ii) The set of natural numbers Z*>*0 with the binary operation of addition.

(b) (**2 marks**) Give an example of a set with a binary operation that is not associative;no explanation is necessary.

A2. (a) (**4 marks**) Suppose *G *is a multiplicatively written group with identity element *e*, and *x, y **∈ **G*. Carefully using the defifinition of a group, show that if *xyx *= *x *2 then *y *= *e*. 数学代写价格

(b) (**4 marks**) Show that *H *= *{*2 *i *: *i **∈ *Z*} *is a subgroup of (R*>*0*, **×*).

#### A3. For *n **∈ *Z*>*0, let *U**n *be the multiplicative group of congruence classes [*a*] *∈ *Z*/n*Z for which hcf(*a, n*) = 1.

(a) (**4 marks**) Show that [17] *∈ **U*40, and fifind its inverse.

(b) (**4 marks**) What is the order of *U*40? Is 175 *≡ *1(mod 40) true or false? Explain.

A4. (a) (**4 marks**) Write the element

(1*, *3*, *2*, *4*, *9*, *8*, *5)(1*, *2)(4*, *8*, *6)(2*, *9*, *3)

of the symmetric group *S*9 as a product of disjoint cycles and compute its order.

(b) (**4 marks**) Is the alternating group *A*4 abelian? Explain.

A5. (a) (**4 marks**) Let *D*_{10}be the dihedral group of order 10, with standard elements *a, b*.

Write *ab*^{2}*a*^{7}*ba ^{4}* in the form

*a*

^{i}*or*

*a*

^{i}

*b*for some 0

*≤*

*i <*5.

(b) (**4 marks**) Let *D*20 be the dihedral group of order 20, and let *C*5 = *h **c**i *be a cyclic group of order 5 with generator *c*. Is the function *f *: *D*20 *→ **C*5 defifined by

*f*(*a^{i}*) =

*c*

*i*

*, f*(

*a*

^{i}

*b*) =

*c*

^{i}a homomorphism? Explain.

**Section B: Longer Questions 数学代写价格**

**Please use a new booklet for this Part. **

B1. (a) (**7+5 marks**) Consider the set of matrices

with binary operation given by matrix multiplication *·*.

(i) Show that (*G, **·*) is an abelian group.

(ii) Show that (*G, **·*) is isomorphic to the group (R 2 *, *+).

(b) (**5 marks**) Find 0 *≤ **a < *23 so that 350 *≡ **a*(mod 23) using Fermat’s little theorem or otherwise.

(c) (**5 marks**) If an element *x *of a group *G *has order 30, what is the order of *x *18?

(d) (**5+3 marks**) Suppose *G, H *are (multiplicatively-written groups) and *x **∈ **G, y **∈ **H *have fifinite order.

(i) Show directly that (*g, h*) *∈ **G **× **H *has fifinite order.

(ii) If *g *and *h *each have order at most 10, what is the largest possible order of (*g, h*)? 数学代写价格

#### B2. (a) (**4+3 marks**) Let R_{≠0}be the group of non-zero real numbers under the operation of multiplication. Defifine *f *: R_{≠0}*→ *R_{≠0 }by *f*(*x*) = *|**x**|*, the absolute value of *x*.

(i) Show *f *is a homomorphism.

(ii) State the consequence of the homomorphism theorem applied to *f*.

(b) (**5+5 marks**) Let *G *= (R*>*0*, **·*) be the group of positive real numbers under multipli-cation, and let *X *= R. Defifine *g **· **x *= *gx *for *g **∈ **G *and *x **∈ **X*, where *gx *is the usual

multiplication of real numbers.

(i) Show that *g **· **x *= *gx *defifines an action of *G *on *X*.

(ii) Describe the orbits and stabilizers of this action.

(c) (**4+4 marks**)

(i) Give an example of a subgroup *H *of a group *G *that is not normal, with a brief explanation.

(ii) Suppose *H *is a subgroup of a group *G *with the property that for any *x **∈ **G*,there exists *y **∈ **G *so that *xH *= *Hy*. Show that *H *is a normal subgroup of *G*.

(d) (**5 marks**) Suppose *G *is a group of order 50 that acts on a set *X *of size 4, with *g **· **x *≠*x *for some *g **∈ **G *and some *x **∈ **X*. Show that *G *must have a subgroup of

order 25.

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