Prior content: None
Previous Section: Number Theory
Next Section: Complex Numbers
Contents
- Whole Topic Notes
- 2A The Axioms for a Group
- 2B Cayley Tables and Finite Groups
- 2C Order and Subgroups
- 2D Isomorphism*
- Whole Topic Summary Resources (Including Past Paper Questions)
Whole Topic Notes
Full Notes to Accompany Examples
Dr Frost
- Dr Frost Further Pure 2 Groups Section
2A The Axioms for a Group
Textbook resources
FMSP Video Courtesy of MEI
Dr Frost resources
No Live Lesson Video Yet
Key Skills
Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost
- 799a: Determine whether a set is closed under a binary operator.
- 799b: Identify an identity element of a set under a binary operator.
- 799c: Show that an operation is associative on a set.
- 799d: Show that all elements of a set have a inverse under a binary operator.
- 799e: Show that a set forms a group under a binary operator.
- 799h: Prove a statement given a group.
- 799: Exam Practice: Axioms of a group
2B Cayley Tables and Finite Groups
Textbook resources
No Video Yet
Dr Frost resources
No Live Lesson Video Yet
Key Skills
Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost
- 800a: Form a Cayley table for a group (excluding modular arithmetic) and understand its key properties.
- 800b: Form a Cayley table for a group modulo n
- 800c: Read off a value from a Cayley table involving an expression with multiple applications of an operator.
- 800e: Show that a collection of permutations form a group under composition.
- 800f: Show that a set of geometric transformations together with composition form a group of symmetries.
- 800g: Show that a value is a generator of a cyclic group.
- 800h: Determine a generator of a cyclic group.
- 800i: Determine whether a group is cyclic.
- 800: Exam Practice: Cayley tables and cyclic groups
2C Order and Subgroups
Textbook resources
No Video Yet
Dr Frost resources
No Live Lesson Video Yet
Key Skills
Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost
- 802a: Identify the order of an element of a group given the set and operator.
- 802b: Identify the order of an element of a group using a Cayley table.
- 800d: Use the associativity of an operator of a group to determine a value in a Cayley table.
- 802c: Identify when no element of a group can have a self-inverse, using Lagrange's theorem.
- 802d: Determine whether a subset forms a subgroup of a group.
- 802e: Identify a subgroup of a group of symmetries.
- 802f: Use Fermat's Little Theorem to determine possible orders of a subgroup.
- 802: Exam Practice: Orders of groups, orders of elements and subgroups
2D Isomorphism*
Textbook resources
No Video Yet
Dr Frost resources
No Live Lesson Video Yet
Key Skills
Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost
Worksheets
- Mathshelper (Jon Stone)
Whole Topic Summary Resources (Including Past Paper Questions)
External whole topic past paper style questions
- WW Dr Frost Style QQQs (Version 1) Word Doc, PDF (Q’s Only) or PDF (Q’s & Sol.)
- WW Dr Frost Style QQQs (Version 2) Word Doc, PDF (Q’s Only) or PDF (Q’s & Sol.)
Activities
- NRICH (student problems)