Group axioms and examplesEdexcel A-Level Further Maths: Mind map

Exam questions on Group axioms and examples

1. 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
2. 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
3. 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