What is a binary operation?

A rule combining two elements of a set to give one element.

What is the inverse of a?

The element a^{-1} with aa^{-1}=a^{-1}a=e.

What is a Cayley table?

A grid of all products of pairs of elements of a finite group.

What is a cyclic group?

A group generated by a single element, so every element is a power of it.

A generator of \{1,\ldots,6\} under multiplication mod 7?

3, since 3,2,6,4,5,1 covers all six elements.

How many symmetries does an equilateral triangle have?

Six: three rotations (including the identity) and three reflections.

Group axioms and examplesEdexcel A-Level Further Maths: Flashcards

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State the four group axioms.

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State the four group axioms.: Closure, associativity, identity, inverses.
What does closure mean?: For all $a,b\in G$ , $a*b\in G$ .
What is a binary operation?: A rule combining two elements of a set to give one element.
What is the inverse of $a$ ?: The element $a^{-1}$ with $a*a^{-1}=a^{-1}*a=e$ .
Inverse of $a\neq0$ in the integers modulo $n$ under addition?: $n-a$ .
When is $\{1,\ldots,n-1\}$ under multiplication mod $n$ a group?: When $n$ is prime.
Why are singular matrices excluded from matrix groups?: They have no inverse.
What is a Cayley table?: A grid of all products of pairs of elements of a finite group.
What must appear in each row and column of a Cayley table?: Every element exactly once.
What is a cyclic group?: A group generated by a single element, so every element is a power of it.
A generator of $\{1,\ldots,6\}$ under multiplication mod 7?: $3$ , since $3,2,6,4,5,1$ covers all six elements.
How many symmetries does an equilateral triangle have?: Six: three rotations (including the identity) and three reflections.

Exam questions on Group axioms and examples

The set $G=\{1,3,5,7\}$ under multiplication modulo 8 is a group.
Determine whether $G$ is a cyclic group, justifying your answer.2 marks
The set $G=\{1,2,3,4,5,6\}$ under multiplication modulo 7 is a group.
Show that the set $\{1,2,3,4,5\}$ under multiplication modulo 6 is not a group.2 marks
Let $e$ be the identity permutation of $\{1,2,3\}$ , let $r$ be the permutation $1\to2,\ 2\to3,\ 3\to1$ , and let $s$ be the permutation that swaps $1$ and $2$ and fixes $3$ . The product $xy$ means apply $y$ first, then $x$ . The six permutations of $\{1,2,3\}$ form a group under this product.
Find $rs$ and $sr$ , writing each as a permutation, and state what this shows about the group.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).