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

Exam questions on Lagrange's theorem and generators

1. $G$ 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
2. $G$ is a group of order $7$.
   Explain why $G$ must be cyclic.2 marks
3. $G=\langle a\rangle$ 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