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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn9FM0Option · Further Pure Mathematics 2GroupsGroups and subgroups

Groups · Subtopic 1 of 1

Groups

A group is a set with an operation that is closed and associative, has an identity, and gives every element an inverse. The order of an element is the smallest power giving the identity. By Lagrange's theorem, the order of any subgroup divides the order of the group.

Edexcel 9FM0 OptionAQA 7367 Paper 3≈ 50 minRevision 1 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Check the four group axioms for a set and binary operation.
Construct and read Cayley tables.
Find orders of elements; recognise cyclic groups and generators.
Use Lagrange's theorem to find possible subgroups; identify isomorphic groups.

Pause and think

Rotations of a square by $0^\circ,90^\circ,180^\circ,270^\circ$ : combining two always gives another. Can every rotation be undone by one of them?

Section 2 of 14

Core idea

Start hereSymmetries you can combine

Turn a square through $90^\circ$ , then $180^\circ$ : you've turned it $270^\circ$ . Any two rotations combine into another, there's a "do nothing" rotation, and every rotation can be undone. Those few facts, and nothing else, are what make something a group.

The big idea

A group $(G,*)$ is a set with an operation that is closed, associative, has an identity $e$ , and gives every element an inverse. From those four axioms follow orders of elements, cyclic groups, subgroups and Lagrange's theorem.

1. The four axioms

a*b\in G;\ (a*b)*c=a*(b*c);\ a*e=e*a=a;\ a*a^{-1}=a^{-1}*a=e

$e$: identity element.
$a^{-1}$: inverse of $a$ .

Worked exampleA non-standard operation

On $\mathbb Z$ , $a*b=a+b-2$ . Show there is an identity and find the inverse of $5$ .

Goal: $e$ and $5^{-1}$ .
Strategy: Solve $a*e=a$ , then $5*x=e$ .

$a+e-2=a\Rightarrow e=2$
Definition of identity.
$5+x-2=2$
Inverse combines to $e=2$ .
$x=-1$
Solve.

Answer

e=2

5^{-1}=-1

✓ Check

5*(-1)=5-1-2=2

✓.

2. Orders and cyclic groups

Powers of

3

under

\times_7

$3^1=3,\ 3^2=9\equiv2$
Reduce mod $7$ each time.
$3^3\equiv6,\ 3^4\equiv18\equiv4$
Multiply the previous result by $3$ .
$3^5\equiv12\equiv5,\ 3^6\equiv15\equiv1$
Back to the identity.

Order

6

: the powers list every element, so

3

is a generator and the group is cyclic.

Why Lagrange's theorem holds (sketch)

$gH=\{gh:h\in H\}$
Form cosets of the subgroup $H$ .
$|gH|=|H|$
$h\mapsto gh$ is one-to-one (cancel $g$ ).
$\text{cosets are equal or disjoint}$
If they share an element they share all.
$|G|=k|H|$
$G$ splits into $k$ equal pieces.

|H|

divides

|G|

, and the order of each element divides

|G|

Worked examplePossible subgroups

A group has order $15$ . What orders can its subgroups have?

Goal: A list.
Strategy: Lagrange.

$|H|\mid15$
Lagrange.
$1,3,5,15$
Divisors of $15$ .

Answer

1,3,5,15

; proper non-trivial subgroups have order

3

5

✓ CheckLagrange says which orders are possible, not that each one exists; for order

15

they do.

Testing for a group

Drawing diagram…

Fail any one test and it is not a group.

Key takeawayIsomorphism Two groups are isomorphic if a one-to-one matching preserves the operation. Quick checks: same order, same number of elements of each order.

Common misconceptionZero under multiplication

\{0,1,2,3\}

under

\times_4

is not a group:

0

and

2

have no inverses.

Pause and think

Can a group of order 7 have a subgroup of order 2?

Summary

Closure, associativity, identity, inverses.
Order of $a$ : least $n$ with $a^n=e$ .
Cyclic: generated by one element.
Lagrange: $|H|$ divides $|G|$ .

Abelian: Group with $ab=ba$ for all elements.
Cayley table: Grid of all products in a group.
Subgroup: Subset that is a group under the same operation.

HistoryÉvariste Galois. Galois invented group theory to show which polynomial equations can be solved by radicals. He wrote it up the night before a fatal duel in 1832, aged 20.

Fun factRubik's cube. The moves of a Rubik's cube form a group with

43\,252\,003\,274\,489\,856\,000

elements, and every position can be solved in 20 moves or fewer.

Section 3 of 14

Key formulae

Axioms

\text{closure},\ (ab)c=a(bc),\ \exists e: ae=ea=a,\ \forall a\ \exists a^{-1}: aa^{-1}=e

Order of an element

\operatorname{ord}(a)=\min\{n\ge1: a^n=e\}

Lagrange's theorem

|H|\ \text{divides}\ |G|\ \text{for every subgroup }H

TrickLatin squares. In a group's Cayley table every element appears exactly once in each row and column. If not, it isn't a group.

Section 4 of 14

Worked examples

Example 1 · Multiplication mod 70 / 3 steps

Find the order of

3

\{1,2,3,4,5,6\}

under

\times_7

Example 2 · A strange operation0 / 2 steps

\mathbb Z

a*b=a+b-2

. Find the identity and the inverse of

5

Section 5 of 14

Explore the graph

Powers of a rotation

Loading the interactive graph…

Try this

Rotation by $\frac{2\pi}n$ has order $n$ : after $n$ steps you're back at the start.
This cyclic group of order $n$ is isomorphic to integers under $+_n$ .

The cyclic group $C_n$ as rotations.

In the real worldCryptography. Online banking uses groups of points on elliptic curves; security rests on how hard it is to undo repeated group operations.

Section 6 of 14

Common mistakes

IncorrectAssuming commutativity

CorrectNot required

Groups need not be abelian (e.g. matrix multiplication).

IncorrectForgetting closure

CorrectCheck results stay in the set

E.g. odd numbers under

+

fail closure.

IncorrectOrder of element

=

the element

CorrectSmallest

n

with

a^n=e

(\mathbb Z_{10},+)

2

has order

5

Me: is it a group?

Closure, identity, inverses, associativity. No vibes allowed.

Section 7 of 14

Exam tip

Exam tipAssociativity from known structures. You can usually quote associativity: addition, multiplication, matrix multiplication and composition of functions are all associative.

Section 8 of 14

Guided practice

SA-FP-GR-G012 marksL2 StandardNon-calculatorOrders

Find the order of

3

in the group

\{1,2,3,4,5,6\}

under multiplication mod

7

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-FP-GR-0011 markL1 FoundationNon-calculatorAxioms

\{1,3,5,7\}

under multiplication mod

8

, find the inverse of

5

Salam Academy Original Practice · Proposed mark scheme

SA-FP-GR-0021 markL2 StandardNon-calculatorOrders

Find the order of

2

(\mathbb Z_{10},+_{10})

Salam Academy Original Practice · Proposed mark scheme

SA-FP-GR-0032 marksL2 StandardNon-calculatorSubgroups

A group has order

12

. How many different orders are possible for its subgroups (including trivial and whole group)?

Salam Academy Original Practice · Proposed mark scheme

SA-FP-GR-0042 marksL2 StandardNon-calculatorOrders

How many generators does the cyclic group

(\mathbb Z_8,+_8)

have?

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-FP-GR-0053 marksL3 ExamNon-calculatorAxioms

\mathbb Z

, define

a*b=a+b-2

. Find the inverse of

5

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Four axioms.
Order of element: first power giving $e$ .
Cyclic: one element generates all.
Lagrange: subgroup orders divide $|G|$ .

Section 12 of 14

Quick quiz

Q1 · SA-FP-GR-Q011 markL1 FoundationNon-calculatorAxioms

Which is NOT one of the group axioms?

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: A level Further Mathematics 9FM0
Where: Further Pure 2 · Groups
Expected: Group axioms, Cayley tables, orders of elements and groups, cyclic groups, subgroups, Lagrange's theorem, isomorphism.
Calculator: Permitted; show enough working to earn the method marks.
Typical style: Edexcel often opens with a routine part and builds to a "show that" or "prove" part.
Open in: 9FM0 · Option

Section 14 of 14

Past paper questions

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Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Pearson Edexcel 9FM0, First teaching 2017 (official specification).