AQA
Further Paper 3 Discrete
2021
June
— Question 5
Exam Board
AQA
Module
Further Paper 3 Discrete (Further Paper 3 Discrete)
Year
2021
Session
June
Topic
Groups
5
Describe the conditions necessary for a set of elements, \(S\), under a binary operation * to form a group.
5
In the multiplicative group of integers modulo 13, the group \(G\) is defined as
$$G = \left( \langle 10 \rangle , \times _ { 13 } \right)$$
5
Explain why \(G\) is an abelian group.
5
(ii) Find the order of \(G\).
5
State the identity element of \(G\) and prove it is an identity element.
Fully justify your answer.
5
Find all the proper non-trivial subgroups of \(G\), giving your answers in the form \(\left( \langle g \rangle , \times _ { 13 } \right)\), where \(g\) is an integer less than 13