OCR MEI Further Extra Pure Specimen — Question 1 10 marks

Exam BoardOCR MEI
ModuleFurther Extra Pure (Further Extra Pure)
SessionSpecimen
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicGroups
TypeOrder of elements and cyclic structure
DifficultyChallenging +1.8 This is a Further Maths group theory question requiring understanding of cyclic groups, generators, and isomorphisms. While the concepts are advanced, the execution is relatively mechanical: computing powers modulo 19, identifying generators by order, and constructing explicit isomorphisms. The multi-part structure and abstract algebra content place it well above typical A-level, but it's a standard textbook exercise for Further Pure rather than requiring novel insight.
Spec8.03c Group definition: recall and use, show structure is/isn't a group8.03h Generators: of cyclic and non-cyclic groups8.03l Isomorphism: determine using informal methods
The set \(G = \{1, 4, 5, 6, 7, 9, 11, 16, 17\}\) is a group of order 9 under the binary operation of multiplication modulo 19.
  1. Show that \(G\) is a cyclic group generated by the element 4. [3]
  2. Find another generator for \(G\). Justify your answer. [2]
  3. Specify two distinct isomorphisms from the group \(J = \{0, 1, 2, 3, 4, 5, 6, 7, 8\}\) under addition modulo 9 to \(G\). [5]
Question 1:
AnswerMarks Guidance
1(i) Order of G is 9, so 4 has order [1,] 3 or 9.
414,43 7
49 73 1
AnswerMarks
So 4 has order 9 and G is a cyclic group generated by 4.E1
M1
AnswerMarks
A12.1
1.1
AnswerMarks
1.1Use of modulo
arithmetic
Alternative Method
AnswerMarks Guidance
n1 2
4n4 16
4 has order 9 and generates G so G is a cyclic group generated by the element 4E1
[3]
AnswerMarks Guidance
1(ii) One of 16, 9, 17, 6, 5
Appropriate reason e.g. 54 1, reference to table if given in (i).B1
E1
AnswerMarks
[2]1.1
2.4
AnswerMarks Guidance
1(iii) In J, elements of order 9 are 1, 2, 4, 5, 7 and 8.
4 must correspond to an element of order 9.
n 1 2 3 4 5 6 7 8 9
4n in G 4 16 7 9 17 11 6 5 1
1 added n times in J 1 2 3 4 5 6 7 8 0 Each row
2 added n times in J 2 4 6 8 1 3 5 7 0 represents
4 added n times in J 4 8 3 7 2 6 1 5 0 a possible
isomorphism
5 added n times in J 5 1 6 2 7 3 8 4 0
with
7 added n times in J 7 5 3 1 8 6 4 2 0
4, 16, 7, …, 1.
8 added n times in J 8 7 6 5 4 3 2 1 0 Two are
AnswerMarks
asked for.M1
E1
M1
A1
A1
AnswerMarks
[5]3.1a
2.4
1.1
1.1
AnswerMarks
2.1For naming two
First isomorphism
has five correct
correspondences
First isomorphism
correct
Second isomorphism
correct
AnswerMarks Guidance
n1 2
4n in G4 16
1 added n times in J1 2
represents
a possible
isomorphism
with
4, 16, 7, …, 1.
Two are
asked for.
AnswerMarks Guidance
2 added n times in J2 4
4 added n times in J4 8
5 added n times in J5 1
7 added n times in J7 5
8 added n times in J8 7 
Question 1:
1 | (i) | Order of G is 9, so 4 has order [1,] 3 or 9.
414,43 7
49 73 1
So 4 has order 9 and G is a cyclic group generated by 4. | E1
M1
A1 | 2.1
1.1
1.1 | Use of modulo
arithmetic
Alternative Method
n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | M1 | Attempt
4n | 4 | 16 | 7 | 9 | 17 | 11 | 6 | 5 | 1 | A1 | Correct
4 has order 9 and generates G so G is a cyclic group generated by the element 4 | E1
[3]
1 | (ii) | One of 16, 9, 17, 6, 5
Appropriate reason e.g. 54 1, reference to table if given in (i). | B1
E1
[2] | 1.1
2.4
1 | (iii) | In J, elements of order 9 are 1, 2, 4, 5, 7 and 8.
4 must correspond to an element of order 9.
n 1 2 3 4 5 6 7 8 9
4n in G 4 16 7 9 17 11 6 5 1
1 added n times in J 1 2 3 4 5 6 7 8 0 Each row
2 added n times in J 2 4 6 8 1 3 5 7 0 represents
4 added n times in J 4 8 3 7 2 6 1 5 0 a possible
isomorphism
5 added n times in J 5 1 6 2 7 3 8 4 0
with
7 added n times in J 7 5 3 1 8 6 4 2 0
4, 16, 7, …, 1.
8 added n times in J 8 7 6 5 4 3 2 1 0 Two are
asked for. | M1
E1
M1
A1
A1
[5] | 3.1a
2.4
1.1
1.1
2.1 | For naming two
First isomorphism
has five correct
correspondences
First isomorphism
correct
Second isomorphism
correct
n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
4n in G | 4 | 16 | 7 | 9 | 17 | 11 | 6 | 5 | 1
1 added n times in J | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 0 | Each row
represents
a possible
isomorphism
with
4, 16, 7, …, 1.
Two are
asked for.
2 added n times in J | 2 | 4 | 6 | 8 | 1 | 3 | 5 | 7 | 0
4 added n times in J | 4 | 8 | 3 | 7 | 2 | 6 | 1 | 5 | 0
5 added n times in J | 5 | 1 | 6 | 2 | 7 | 3 | 8 | 4 | 0
7 added n times in J | 7 | 5 | 3 | 1 | 8 | 6 | 4 | 2 | 0
8 added n times in J | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0
The set $G = \{1, 4, 5, 6, 7, 9, 11, 16, 17\}$ is a group of order 9 under the binary operation of multiplication modulo 19.

\begin{enumerate}[label=(\roman*)]
\item Show that $G$ is a cyclic group generated by the element 4. [3]

\item Find another generator for $G$. Justify your answer. [2]

\item Specify two distinct isomorphisms from the group $J = \{0, 1, 2, 3, 4, 5, 6, 7, 8\}$ under addition modulo 9 to $G$. [5]
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Extra Pure  Q1 [10]}}
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