What is the order of an element g?

The smallest positive integer n with g^n=e.

A subset of a group that is itself a group under the same operation.

State Lagrange's theorem.

The order of a subgroup of a finite group divides the order of the group.

How does Lagrange's theorem restrict element orders?

The order of every element divides the order of the group.

What can be said about a group of prime order?

It is cyclic, and its only subgroups are the trivial ones.

How do you show a finite group is cyclic using orders?

Find an element whose order equals the order of the group.

Does every divisor of |G| occur as a subgroup order?

Not necessarily: Lagrange's theorem only says the order must be a divisor.

Order, subgroups and Lagrange's theoremEdexcel A-Level Further Maths: Flashcards

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What is the order of a group?

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What is the order of a group?: The number of elements in it.
What is the order of an element $g$ ?: The smallest positive integer $n$ with $g^n=e$ .
What is a subgroup?: A subset of a group that is itself a group under the same operation.
What are the trivial subgroups?: $\{e\}$ and $G$ itself.
State Lagrange's theorem.: The order of a subgroup of a finite group divides the order of the group.
What is the order of the cyclic subgroup $\langle g\rangle$ ?: The order of $g$ .
How does Lagrange's theorem restrict element orders?: The order of every element divides the order of the group.
What can be said about a group of prime order?: It is cyclic, and its only subgroups are the trivial ones.
How do you show a finite group is cyclic using orders?: Find an element whose order equals the order of the group.
Can a group of order 12 have a subgroup of order 8?: No: 8 does not divide 12.
Does every divisor of $|G|$ occur as a subgroup order?: Not necessarily: Lagrange's theorem only says the order must be a divisor.
Order of $3$ in $\{0,\ldots,11\}$ under addition mod 12?: $4$ ( $3,6,9,0$ ).

Exam questions on Order, subgroups and Lagrange's theorem

The set $G=\{1,2,3,\ldots,10\}$ under multiplication modulo 11 is a group.
Find a subgroup of $G$ of order $5$ , listing its elements.2 marks
$G$ is a finite group of order $24$ .
An element $g$ of $G$ has order $24$ . Explain why $G$ must be cyclic.2 marks
The symmetries of an equilateral triangle form a group $T$ of order $6$ under composition. Its elements are the identity $e$ , a rotation $r$ through $120^\circ$ , the rotation $r^2$ through $240^\circ$ , and three reflections $m_1,m_2,m_3$ .
State, with justification, the orders of $r$ , $r^2$ and $m_1$ .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).