Lagrange's theorem and generatorsAQA A-Level Further Maths: Flashcards

What these 12 flashcards ask

State Lagrange's theorem.
Possible orders of subgroups of a group of order 12?
Can a group of order 12 have a subgroup of order 8?
What does Lagrange say about the order of an element?
What is a generator?
What is \langle a\rangle?
Why is every group of prime order cyclic?
Subgroups of a group of prime order p?
Order of a^k in a cyclic group of order n?
When is a^k a generator of a cyclic group of order n?
Generators of a cyclic group of order 10 generated by g?
Does Lagrange guarantee a subgroup of every divisor order?

Exam questions on Lagrange's theorem and generators

$G$ is a group of order $12$ .
$G$ contains an element $g$ of order $6$ . $K$ is a subgroup of $G$ that contains $g$ . Find the possible orders of $K$ .2 marks
$G$ is a group of order $7$ .
Explain why $G$ must be cyclic.2 marks
$G=\langle a\rangle$ is a cyclic group of order $18$ , with identity $e$ , so $a^{18}=e$ .
Find the order of each of $a^4$ , $a^6$ and $a^9$ .3 marks

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).