| Date | November 2016 | Marks available | 8 | Reference code | 16N.3srg.hl.TZ0.3 |
| Level | HL only | Paper | Paper 3 Sets, relations and groups | Time zone | TZ0 |
| Command term | State | Question number | 3 | Adapted from | N/A |
Question
An Abelian group, \(\{ G,{\text{ }} * \} \), has 12 different elements which are of the form \({a^i} * {b^j}\) where \(i \in \{ 1,{\text{ }}2,{\text{ }}3,{\text{ }}4\} \) and \(j \in \{ 1,{\text{ }}2,{\text{ }}3\} \). The elements \(a\) and \(b\) satisfy \({a^4} = e\) and \({b^3} = e\) where \(e\) is the identity.
Let \(\{ H,{\text{ }} * \} \) be the proper subgroup of \(\{ G,{\text{ }} * \} \) having the maximum possible order.
State the possible orders of an element of \(\{ G,{\text{ }} * \} \) and for each order give an example of an element of that order.
(i) State a generator for \(\{ H,{\text{ }} * \} \).
(ii) Write down the elements of \(\{ H,{\text{ }} * \} \).
(iii) Write down the elements of the coset of \(H\) containing \(a\).
Markscheme
orders are 1 2 3 4 6 12 A2
Note: A1 for four or five correct orders.
Note: For the rest of this question condone absence of xxx and accept equivalent expressions.
\(\begin{array}{*{20}{l}} {{\text{order:}}}&1&{{\text{element:}}}&2&{A1} \\ {}&2&{}&{{a^2}}&{A1} \\ {}&3&{}&{b{\text{ or }}{{\text{b}}^2}}&{A1} \\ {}&4&{}&{a{\text{ or }}{a^3}}&{A1} \\ {}&6&{}&{{a^2} * b{\text{ or }}{a^2} * {b^2}}&{A1} \\ {}&{12}&{}&{a * b{\text{ or }}a * {b^2}{\text{ or }}{a^3} * b{\text{ or }}{a^3} * {b^2}}&{A1} \end{array}\)
[8 marks]
(i) \(H\) has order 6 (R1)
generator is \({a^2} * b\) or \({a^2} * {b^2}\) A1
(ii) \(H = \left\{ {e,{\text{ }}{a^2} * b,{\text{ }}{b^2},{\text{ }}{a^2},{\text{ }}b,{\text{ }}{a^2} * {b^2}} \right\}\) A3
Note: A2 for 4 or 5 correct. A1 for 2 or 3 correct.
(iii) required coset is \(Ha\) (or \(aH\)) (R1)
\(Ha = \left\{ {a,{\text{ }}{a^3} * b,{\text{ }}a * {b^2},{\text{ }}{a^3},{\text{ }}a * b,{\text{ }}{a^3} * {b^2}} \right\}\) A1
[7 marks]
