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</div><h2>SL Paper 2</h2><div class="specification">
<p>The set of all integer s from 0 to 99 inclusive is denoted by <em>S</em>. The binary operations \( * \) and \( \circ \) are defined on <em>S</em> by</p>
<p style="padding-left: 180px;">\(a * b = \left[ {a + b + 20} \right]\)(mod 100)</p>
<p style="padding-left: 180px;">\(a \circ b = \left[ {a + b - 20} \right]\)(mod 100).</p>
</div>
<div class="specification">
<p>The equivalence relation <em>R</em> is defined by \(aRb \Leftrightarrow \left( {{\text{sin}}\frac{{\pi a}}{5} = {\text{sin}}\frac{{\pi b}}{5}} \right)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the identity element of <em>S</em> with respect to \( * \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that every element of <em>S</em> has an inverse with respect to \( * \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State which elements of <em>S</em> are self-inverse with respect to \( * \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove that the operation \( \circ \) is not distributive over \( * \).</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the equivalence classes into which <em>R</em> partitions <em>S</em>, giving the first four elements of each class.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find two elements in the same equivalence class which are inverses of each other with respect to \( * \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the set \(J = \left\{ {a + b\sqrt 2 :a,{\text{ }}b \in \mathbb{Z}} \right\}\) under the binary operation multiplication.</p>
</div>
<div class="specification">
<p class="p1">Consider \(a + b\sqrt 2 \in G\)<span class="s1">, where \(\gcd (a,{\text{ }}b) = 1\),</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(J\) <span class="s1">is closed.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State the identity in \(J\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that</p>
<p class="p1">(i) <span class="Apple-converted-space"> \(1 - \sqrt 2 \)</span> has an inverse in \(J\)<span class="s1">;</span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> \(2 + 4\sqrt 2 \)</span> has no inverse in \(J\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Show that the subset, \(G\)</span>, of elements of <span class="s1">\(J\) </span>which have inverses, forms a group of infinite order.</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the inverse of \(a + b\sqrt 2 \)<span class="s1">.</span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Hence show that \({a^2} - 2{b^2}\) divides exactly into \(a\) and \(b\).</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Deduce that \({a^2} - 2{b^2} = \pm 1\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Draw the Cayley table for the set \(S = \left\{ {0,1,2,3,4,\left. 5 \right\}} \right.\) under addition </span><span style="font-family: times new roman,times; font-size: medium;">modulo six \(({ + _6})\) and hence show that \(\left\{ {S, + \left. {_6} \right\}} \right.\) is a group.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Show that the group is cyclic and write down its generators.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) Find the subgroup of \(\left\{ {S, + \left. {_6} \right\}} \right.\) that contains exactly three elements.</span></p>
<div class="marks">[11]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Prove that a cyclic group with exactly one generator cannot have more than </span><span style="font-family: times new roman,times; font-size: medium;">two elements.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(H\) is a group and the function \(\Phi :H \to H\) is defined by \(\Phi (a) = {a^{ - 1}}\) , where \({a^{ - 1}}\) </span><span style="font-family: times new roman,times; font-size: medium;">is the inverse of <strong><em>a</em></strong> under the group operation. Show that \(\Phi \) is an isomorphism </span><span style="font-family: times new roman,times; font-size: medium;"><strong>if and only if</strong> <em><strong>H</strong></em> is Abelian.</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The function \(f:\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}\)</span><span style="font-family: times new roman,times; font-size: medium;"> is defined by \(\boldsymbol{X} \mapsto \boldsymbol{AX}\) , where \(\boldsymbol{X} = \left[ \begin{array}{l}<br>x\\<br>y<br>\end{array} \right]\) and \(\boldsymbol{A} = \left[ \begin{array}{l}<br>a\\<br>c<br>\end{array} \right.\left. \begin{array}{l}<br>b\\<br>d<br>\end{array} \right]\) </span><span style="font-family: times new roman,times; font-size: medium;">where \(a\) , \(b\) , \(c\) , \(d\) are all non-zero.</span></p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider the group \(\left\{ {S,{ + _m}} \right\}\) where \(S = \left\{ {0,1,2 \ldots m - 1} \right\}\) , \(m \in \mathbb{N}\) , \(m \ge 3\) and \({ + _m}\) </span><span style="font-family: times new roman,times; font-size: medium;">denotes addition modulo \(m\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(f\) is a bijection if \(\boldsymbol{A}\) is non-singular.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">A.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Suppose now that \(\boldsymbol{A}\) is singular.</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) Write down the relationship between \(a\) , \(b\) , \(c\) , \(d\) .</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Deduce that the second row of \(\boldsymbol{A}\) is a multiple of the first row of \(\boldsymbol{A}\) .</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (iii) Hence show that \(f\) is not a bijection.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">A.b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(\left\{ {S,{ + _m}} \right\}\) is cyclic for all <strong><em>m</em></strong> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">B.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that \(m\) is prime,</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) explain why all elements except the identity are generators of \(\left\{ {S,{ + _m}} \right\}\) ;</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (ii) find the inverse of \(x\) , where <strong><em>x</em></strong> is any element of \(\left\{ {S,{ + _m}} \right\}\) apart from the </span><span style="font-family: times new roman,times; font-size: medium;">identity;</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (iii) determine the number of sets of two distinct elements where each element </span><span style="font-family: times new roman,times; font-size: medium;">is the inverse of the other.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">B.b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Suppose now that \(m = ab\) where \(a\) , \(b\) are unequal prime numbers. Show that </span><span style="font-family: times new roman,times; font-size: medium;">\(\left\{ {S,{ + _m}} \right\}\) has two proper subgroups and identify them.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">B.c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The binary operator \( * \) is defined for <strong><em>a</em></strong> , \(b \in \mathbb{R}\) by \(a * b = a + b - ab\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Show that \( * \) is associative.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the identity element.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) Find the inverse of \(a \in \mathbb{R}\) , showing that the inverse exists for all values </span><span style="font-family: times new roman,times; font-size: medium;">of \(a\) except one value which should be identified.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iv) Solve the equation \(x * x = 1\) .</span></p>
<div class="marks">[15]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The domain of \( * \) is now reduced to \(S = \left\{ {0,2,3,4,5,\left. 6 \right\}} \right.\) and the arithmetic is </span><span style="font-family: times new roman,times; font-size: medium;">carried out modulo \(7\).</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (i) Copy and complete the following Cayley table for \(\left\{ {S,\left. * \right\}} \right.\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="images/disco.png" alt></span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Show that \(\left\{ {S,\left. * \right\}} \right.\) is a group.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (iii) Determine the order of each element in <strong><em>S</em></strong> and state, with a reason, </span><span style="font-family: times new roman,times; font-size: medium;">whether or not \(\left\{ {S,\left. * \right\}} \right.\) is cyclic.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (iv) Determine all the proper subgroups of \(\left\{ {S,\left. * \right\}} \right.\) and explain how your results </span><span style="font-family: times new roman,times; font-size: medium;">illustrate Lagrange’s theorem.</span></p>
<p style="margin-left: 30px;"><span style="font-size: medium;"><span style="font-family: times new roman,times;"> (v) Solve the equation</span> <span style="font-family: times new roman,times;">\(2 * x * x = 5\) .</span></span></p>
<div class="marks">[17]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The set \(S\) consists of real numbers <strong><em>r</em></strong> of the form \(r = a + b\sqrt 2 \) , where \(a,b \in \mathbb{Z}\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The relation \(R\) is defined on \(S\) by \({r_1}R{r_2}\) if and only if \({a_1} \equiv {a_2}\) (mod2) and </span><span style="font-family: times new roman,times; font-size: medium;">\({b_1} \equiv {b_2}\) (mod3), where \({r_1} = {a_1} + {b_1}\sqrt 2 \) and \({r_2} = {a_2} + {b_2}\sqrt 2 \) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(R\) is an equivalence relation.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show, by giving a counter-example, that the statement \({r_1}R{r_2} \Rightarrow r_1^2Rr_2^2\) is false.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Determine</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) the equivalence class \(E\) containing \(1 + \sqrt 2 \) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) the equivalence class \(F\) containing \(1 - \sqrt 2 \) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Show that</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \({(1 + \sqrt 2 )^3} \in F\)</span><span style="font-family: times new roman,times; font-size: medium;"> ;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) \({(1 + \sqrt 2 )^6} \in E\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Determine whether the set \(E\) forms a group under</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) the operation of addition;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) the operation of multiplication.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>The set \({S_n} = \{ 1,{\text{ }}2,{\text{ }}3,{\text{ }} \ldots ,{\text{ }}n - 2,{\text{ }}n - 1\} \), where \(n\) is a prime number greater than 2, and \({ \times _n}\) denotes multiplication modulo \(n\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that there are no elements \(a,{\text{ }}b \in {S_n}\) such that \(a{ \times _n}b = 0\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that, for \(a,{\text{ }}b,{\text{ }}c \in {S_n},{\text{ }}a{ \times _n}b = a{ \times _n}c \Rightarrow b = c\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \({G_n} = \{ {S_n},{\text{ }}{ \times _n}\} \) is a group. You may assume that \({ \times _n}\) is associative.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the order of the element \((n - 1)\) is 2.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the inverse of the element 2 is \(\frac{1}{2}(n + 1)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why the inverse of the element 3 is \(\frac{1}{3}(n + 1)\) for some values of \(n\) but not for other values of \(n\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the inverse of the element 3 in \({G_{11}}\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.iv.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the inverse of the element 3 in \({G_{31}}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.v.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The set of all permutations of the list of the integers \(1,{\text{ }}2,{\text{ }}3{\text{ }} \ldots {\text{ }}n\) is a group, \({S_n}\), under the operation of composition of permutations.</p>
</div>
<div class="specification">
<p class="p1"><span class="s1">Each element of \({S_4}\) </span>can be represented by a \(4 \times 4\) matrix. For example, the cycle \({\text{(1 2 3 4)}}\) is represented by the matrix</p>
<p class="p1">\(\left( {\begin{array}{*{20}{c}} 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \\ 1&0&0&0 \end{array}} \right)\) acting on the column vector \(\left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ 3 \\ 4 \end{array}} \right)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">(i) <span class="Apple-converted-space"> </span></span>Show that the order of \({S_n}\) is \(n!\);</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>List the <span class="s2">6 </span>elements of \({S_3}\) <span class="s1">in cycle form;</span></p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Show that \({S_3}\) <span class="s1">is not Abelian;</span></p>
<p class="p1">(iv) <span class="Apple-converted-space"> </span>Deduce that \({S_n}\) is not Abelian for \(n \geqslant 3\).</p>
<div class="marks">[9]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Write down the matrices <span class="s1"><strong><em>M</em></strong>\(_1\), <strong><em>M</em></strong>\(_2\)</span> representing the permutations \((1{\text{ }}2),{\text{ }}(2{\text{ }}3)\)<span class="s1">, </span><span class="s3">respectively;</span></p>
<p class="p2"><span class="s4">(ii) <span class="Apple-converted-space"> </span>Find </span><span class="s1"><strong><em>M</em></strong>\(_1\)<strong><em>M</em></strong>\(_2\)</span> and state the permutation represented by this matrix;</p>
<p class="p1"><span class="s3">(</span>iii) <span class="Apple-converted-space"> </span>Find \(\det (\)<span class="s1"><strong><em>M</em></strong></span><span class="s5">\(_1)\)</span><span class="s1">, \(\det (\)<strong><em>M</em></strong></span><span class="s5">\(_2)\) </span>and deduce the value of \(\det (\)<span class="s1"><strong><em>M</em></strong>\(_1\)<strong><em>M</em></strong></span>\(_2)\)<span class="s5">.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">(i) <span class="Apple-converted-space"> </span></span>Use mathematical induction to prove that</p>
<p class="p1"><span class="s1">\((1{\text{ }}n)(1{\text{ }}n{\text{ }} - 1)(1{\text{ }}n - 2) \ldots (1{\text{ }}2) = (1{\text{ }}2{\text{ }}3 \ldots n){\text{ }}n \in {\mathbb{Z}^ + },{\text{ }}n > 1\).</span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Deduce that every permutation can be written as a product of cycles of length 2.</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f\) be a homomorphism of a group \(G\) onto a group \(H\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that if \(e\) is the identity in \(G\), then \(f(e)\) is the identity in \(H\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that if \(x\) is an element of \(G\), then \(f({x^{ - 1}}) = {\left( {f(x)} \right)^{ - 1}}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that if \(G\) is Abelian, then \(H\) must also be Abelian.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that if \(S\) is a subgroup of \(G\), then \(f(S)\) is a subgroup of \(H\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the special case in which \(G = \{ 1,{\text{ }}3,{\text{ }}4,{\text{ }}9,{\text{ }}10,{\text{ }}12\} ,{\text{ }}H = \{ 1,{\text{ }}12\} \) and \( * \) denotes multiplication modulo 13.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The group \(\{ G,{\text{ }} * \} \) has a subgroup \(\{ H,{\text{ }} * \} \). The relation \(R\) is defined such that for \(x\), \(y \in G\), \(xRy\) if and only if \({x^{ - 1}} * y \in H\). Show that \(R\) is an equivalence relation.</p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that 3\(R\)10.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the three equivalence classes.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">\(S\) is defined as the set of all \(2 \times 2\) <span class="s1">non-singular matrices. </span>\(A\) <span class="s1">and </span>\(B\) <span class="s1">are two elements of the set </span>\(S\)<span class="s1">.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \({({A^T})^{ - 1}} = {({A^{ - 1}})^T}\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Show that \({(AB)^T} = {B^T}{A^T}\).</p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A relation \(R\) is defined on \(S\) such that \(A\) is related to \(B\) if and only if there exists an element \(X\) of \(S\) such that \(XA{X^T} = B\). Show that \(R\) is an equivalence relation.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A group has exactly three elements, the identity element \(e\) , \(h\) and \(k\) . Given the </span><span style="font-family: times new roman,times; font-size: medium;">operation is denoted by \( \otimes \) , show that</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Show that \({\mathbb{Z}_4}\) (the set of integers modulo 4) together with the operation </span><span style="font-family: times new roman,times; font-size: medium;">\({ + _4}\) (addition modulo 4) form a group \(G\) . You may assume associativity.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Show that \(G\) is cyclic.</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">A.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Using Cayley tables or otherwise, show that \(G\) and \(H = \left( {\left\{ {1,2,3,\left. 4 \right\},{ \times _5}} \right.} \right)\) are </span><span style="font-family: times new roman,times; font-size: medium;">isomorphic where \({{ \times _5}}\) is multiplication modulo 5. State clearly all the possible </span><span style="font-family: times new roman,times; font-size: medium;">bijections.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">A.b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">the group is cyclic.<br></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">B.b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">the group is cyclic.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The relation \({R_1}\) is defined for \(a,b \in {\mathbb{Z}^ + }\) by \(a{R_1}b\) if and only if \(n\left| {({a^2} - {b^2})} \right.\) where \(n\) </span><span style="font-family: times new roman,times; font-size: medium;">is a fixed positive integer.</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) Show that \({R_1}\) is an equivalence relation.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Determine the equivalence classes when \(n = 8\) .</span></p>
<div class="marks">[11]</div>
<div class="question_part_label">A.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the group \(\left\{ {G, * } \right\}\) and let \(H\) be a subset of \(G\) defined by</span></p>
<p style="text-align: center;"><span style="font-family: times new roman,times; font-size: medium;">\(H = \left\{ {x \in G} \right.\) such that \(x * a = a * x\) for all \(a \in \left. G \right\}\) .<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(\left\{ {H, * } \right\}\) is a subgroup of \(\left\{ {G, * } \right\}\) .</span></p>
<div class="marks">[12]</div>
<div class="question_part_label">B.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The relation \({R_2}\) is defined for \(a,b \in {\mathbb{Z}^ + }\) by \(a{R_2}b\) if and only if \((4 + \left| {a - b} \right|)\) is the </span><span style="font-family: times new roman,times; font-size: medium;">square of a positive integer. Show that \({R_2}\) is not transitive.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">B.b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The relation \(R\) is defined on \({\mathbb{R}^ + } \times {\mathbb{R}^ + }\) such that \(({x_1},{y_1})R({x_2},{y_2})\) if and only if \(\frac{{{x_1}}}{{{x_2}}} = \frac{{{y_2}}}{{{y_1}}}\) </span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(R\) is an equivalence relation.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Determine the equivalence class containing \(({x_1},{y_1})\) and interpret it geometrically.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The set \(S\) contains the eighth roots of unity given by \(\left\{ {{\text{cis}}\left( {\frac{{n\pi }}{4}} \right),{\text{ }}n \in \mathbb{N},{\text{ }}0 \leqslant n \leqslant 7} \right\}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that \(\{ S,{\text{ }} \times \} \) is a group where \( \times \) denotes multiplication of complex numbers.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Giving a reason, state whether or not \(\{ S,{\text{ }} \times \} \) is cyclic.</span></p>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The binary operation multiplication modulo \(9\), denoted by \({ \times _9}\) , is defined on the set </span><span style="font-family: times new roman,times; font-size: medium;">\(S = \left\{ {1,2,3,4,5,6,7,8} \right\}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Copy and complete the following Cayley table.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/cal.png" alt></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(\left\{ {S,{ \times _9}} \right\}\) is not a group.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Prove that a group \(\left\{ {G,{ \times _9}} \right\}\) can be formed by removing two elements from the </span><span style="font-family: times new roman,times; font-size: medium;">set \(S\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the order of all the elements of \(G\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down all the proper subgroups of \(\left\{ {G,{ \times _9}} \right\}\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) Determine the coset containing the element \(5\) for each of the subgroups in </span><span style="font-family: times new roman,times; font-size: medium;">part (ii).</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Solve the equation \(4{ \times _9}x{ \times _9}x = 1\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The relation \(R\) is defined for \(x,y \in {\mathbb{Z}^ + }\) such that \(xRy\) if and only if </span><span style="font-family: times new roman,times; font-size: medium;">\({3^x} \equiv {3^y}(\bmod 10)\) .</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) Show that \(R\) is an equivalence relation.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Identify all the equivalence classes.</span></p>
<div class="marks">[11]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(S\) denote the set \(\left\{ {x\left| {x = a + b\sqrt 3 ,a,b \in \mathbb{Q},{a^2} + {b^2} \ne 0} \right.} \right\}\) .</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) Prove that \(S\) is a group under multiplication.</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Give a reason why \(S\) would not be a group if the conditions on \({a,b}\) were </span><span style="font-family: times new roman,times; font-size: medium;">changed to \({a,b \in \mathbb{R},{a^2} + {b^2} \ne 0}\) .</span></p>
<div class="marks">[15]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br>