8.03i - OCR Spec

OCR MEI FP3 2014 June Q4

24 marks Challenging +1.8

4 The twelve distinct elements of an abelian multiplicative group $G$ are $$e , a , a ^ { 2 } , a ^ { 3 } , a ^ { 4 } , a ^ { 5 } , b , a b , a ^ { 2 } b , a ^ { 3 } b , a ^ { 4 } b , a ^ { 5 } b$$ where $e$ is the identity element, $a ^ { 6 } = e$ and $b ^ { 2 } = e$.

Show that the element $a ^ { 2 } b$ has order 6 .
Show that $\left\{ e , a ^ { 3 } , b , a ^ { 3 } b \right\}$ is a subgroup of $G$.
List all the cyclic subgroups of $G$. You are given that the set $$H = \{ 1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,59,61,67,71,73,77,79,83,89 \}$$ with binary operation multiplication modulo 90 is a group.
Determine the order of each of the elements 11, 17 and 19 .
Give a cyclic subgroup of $H$ with order 4.
By identifying possible values for the elements $a$ and $b$ above, or otherwise, give one example of each of the following:
(A) a non-cyclic subgroup of $H$ with order 12,
(B) a non-cyclic subgroup of $H$ with order 4.

OCR MEI FP3 2009 June Q4

24 marks Challenging +1.2

4 The group $G = \{ 1,2,3,4,5,6 \}$ has multiplication modulo 7 as its operation. The group $H = \{ 1,5,7,11,13,17 \}$ has multiplication modulo 18 as its operation.

Show that the groups $G$ and $H$ are both cyclic.
List all the proper subgroups of $G$.
Specify an isomorphism between $G$ and $H$. The group $S = \{ \mathrm { a } , \mathrm { b } , \mathrm { c } , \mathrm { d } , \mathrm { e } , \mathrm { f } \}$ consists of functions with domain $\{ 1,2,3 \}$ given by $$\begin{array} { l l l } \mathrm { a } ( 1 ) = 2 & \mathrm { a } ( 2 ) = 3 & \mathrm { a } ( 3 ) = 1 \\ \mathrm {~b} ( 1 ) = 3 & \mathrm {~b} ( 2 ) = 1 & \mathrm {~b} ( 3 ) = 2 \\ \mathrm { c } ( 1 ) = 1 & \mathrm { c } ( 2 ) = 3 & \mathrm { c } ( 3 ) = 2 \\ \mathrm {~d} ( 1 ) = 3 & \mathrm {~d} ( 2 ) = 2 & \mathrm {~d} ( 3 ) = 1 \\ \mathrm { e } ( 1 ) = 1 & \mathrm { e } ( 2 ) = 2 & \mathrm { e } ( 3 ) = 3 \\ \mathrm { f } ( 1 ) = 2 & \mathrm { f } ( 2 ) = 1 & \mathrm { f } ( 3 ) = 3 \end{array}$$ and the group operation is composition of functions.
Show that ad $= \mathrm { c }$ and find da.
Give a reason why $S$ is not isomorphic to $G$.
Find the order of each element of $S$.
List all the proper subgroups of $S$.

OCR MEI FP3 2007 June Q4

24 marks Challenging +1.8

4

Prove that, for a group of order 10, every proper subgroup must be cyclic. The set $M = \{ 1,2,3,4,5,6,7,8,9,10 \}$ is a group under the binary operation of multiplication modulo 11.
Show that $M$ is cyclic.
List all the proper subgroups of $M$. The group $P$ of symmetries of a regular pentagon consists of 10 transformations $$\{ \mathrm { A } , \mathrm {~B} , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm {~F} , \mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm {~J} \}$$ and the binary operation is composition of transformations. The composition table for $P$ is given below.
A B C D E F G H I J
A C J G H A B I F E D
B F E H G B A D C J I
C G D I F C J E B A H
D J C B E D G F I H A
E A B C D E F G H I J
F H I D C F E J A B G
G I H E B G D A J C F
H D G J A H I B E F C
I E F A J I H C D G B
J B A F I J C H G D E
One of these transformations is the identity transformation, some are rotations and the rest are reflections.
Identify which transformation is the identity, which are rotations and which are reflections.
State, giving a reason, whether $P$ is isomorphic to $M$.
Find the order of each element of $P$.
List all the proper subgroups of $P$.

OCR Further Additional Pure 2024 June Q8

15 marks Challenging +1.8

8 The group $G$ is cyclic and of order 12.

1. State the possible orders of all the proper subgroups of $G$. You must justify your answers.
2. List all the elements of each of these subgroups.
3. Explain why $G$ must be abelian. The group $\mathbb { Z } _ { k }$ is the cyclic group of order $k$, consisting of the elements $\{ 0,1,2 , \ldots , k - 1 \}$ under the operation $+ _ { k }$ of addition modulo $k$. The coordinate group $\mathrm { C } _ { \mathrm { mn } }$ is the group which consists of elements of the form $( x , y )$, where $\mathrm { x } \in \mathbb { Z } _ { \mathrm { m } }$ and $\mathrm { y } \in \mathbb { Z } _ { \mathrm { n } }$, under the operation $\oplus$ given by $\left( \mathrm { x } _ { 1 } , \mathrm { y } _ { 1 } \right) \oplus \left( \mathrm { x } _ { 2 } , \mathrm { y } _ { 2 } \right) = \left( \mathrm { x } _ { 1 } + { } _ { \mathrm { m } } \mathrm { x } _ { 2 } , \mathrm { y } _ { 1 } + { } _ { \mathrm { n } } \mathrm { y } _ { 2 } \right)$. For example, for $m = 5$ and $n = 2 , ( 3,0 ) \oplus ( 4,1 ) = ( 2,1 )$.
1. List all the elements of $\mathrm { J } = \mathrm { C } _ { 34 }$.
2. Show that $G$ and $J$ are isomorphic. There is a second coordinate group of order 12; that is, $\mathrm { K } = \mathrm { C } _ { \mathrm { mn } }$, where $1 < \mathrm { m } < \mathrm { n } < 12$ but neither $m$ nor $n$ is equal to 3 or 4 .
1. State the values of $m$ and $n$ which give $K$.
2. Hence list all of the elements of $K$.
3. Explain why $K$ must be abelian.
Show that $G$ and $K$ are not isomorphic. \section*{END OF QUESTION PAPER}

OCR Further Additional Pure Specimen Q8

13 marks Challenging +1.8

8 The set $X$ consists of all $2 \times 2$ matrices of the form $\left( \begin{array} { r r } x & - y \\ y & x \end{array} \right)$, where $x$ and $y$ are real numbers which are not both zero.

(a) The matrices $\left( \begin{array} { c c } a & - b \\ b & a \end{array} \right)$ and $\left( \begin{array} { c c } c & - d \\ d & c \end{array} \right)$ are both elements of $X$. Show that $\left( \begin{array} { c c } a & - b \\ b & a \end{array} \right) \left( \begin{array} { c c } c & - d \\ d & c \end{array} \right) = \left( \begin{array} { c c } p & - q \\ q & p \end{array} \right)$ for some real numbers $p$ and $q$ to be found in terms of $a , b , c$ and $d$.
(b) Prove by contradiction that $p$ and $q$ are not both zero.
Prove that $X$, under matrix multiplication, forms a group $G$. [You may use the result that matrix multiplication is associative.]
Determine a subgroup of $G$ of order 17.

OCR Further Additional Pure 2018 March Q7

14 marks Challenging +1.8

7 The set $M$ contains all matrices of the form $\mathbf { X } ^ { n }$, where $\mathbf { X } = \frac { 1 } { \sqrt { 3 } } \left( \begin{array} { r r } 2 & - 1 \\ 1 & 1 \end{array} \right)$ and $n$ is a positive integer.

Show that $M$ contains exactly 12 elements.
Deduce that $M$, together with the operation of matrix multiplication, form a cyclic group $G$.
Determine all the proper subgroups of $G$. \section*{END OF QUESTION PAPER}

Pre-U Pre-U 9795/1 Specimen Q7

9 marks Challenging +1.8

7 The multiplicative group $G$ has eight elements $e , a , b , c , a b , a c , b c , a b c$, where $e$ is the identity. The group is commutative, and the order of each of the elements $a , b , c$ is 2 .

Find four subgroups of $G$ of order 4.
Give a reason why no group of order 8 can have a subgroup of order 3 . The group $H$ has elements $0,1,2 , \ldots , 7$ with group operation addition modulo 8 .
Find the order of each element of $H$.
Determine whether $G$ and $H$ are isomorphic and justify your conclusion.

OCR FP3 Q1

6 marks Standard +0.3

A group $G$ of order 6 has the combination table shown below.

	$e$	$a$	$b$	$p$	$q$	$r$
$e$	$e$	$a$	$b$	$p$	$q$	$r$
$a$	$a$	$b$	$e$	$r$	$p$	$q$
$b$	$b$	$e$	$a$	$q$	$r$	$p$
$p$	$p$	$q$	$r$	$e$	$a$	$b$
$q$	$q$	$r$	$p$	$b$	$e$	$a$
$r$	$r$	$p$	$q$	$a$	$b$	$e$

State, with a reason, whether or not $G$ is commutative. [1]
State the number of subgroups of $G$ which are of order 2. [1]
List the elements of the subgroup of $G$ which is of order 3. [1]

A multiplicative group $H$ of order 6 has elements $e, c, c^2, c^3, c^4, c^5$, where $e$ is the identity. Write down the order of each of the elements $c^3, c^4$ and $c^5$. [3]

OCR FP3 2008 January Q1