Group axioms and examplesEdexcel A-Level Further Maths: Subtopic test

1. 1
   The set $G=\{1,3,5,7\}$ under multiplication modulo 8 is a group.

   (a)

   What is the inverse of $5$ in $G$?

   [1 mark]

   • A$5$
   • B$3$
   • C$7$
   • D$1$

   (b)

   How many times does the identity element appear in the Cayley table of $G$?

   [1 mark]

   • A$1$
   • B$2$
   • C$3$
   • D$4$

   (c)

   Determine whether $G$ is a cyclic group, justifying your answer.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The set $G=\{1,2,3,4,5,6\}$ under multiplication modulo 7 is a group.

   (a)

   What is the inverse of $3$ in $G$?

   [1 mark]

   • A$2$
   • B$5$
   • C$4$
   • D$6$

   (b)

   Which of these elements generates $G$?

   [1 mark]

   • A$2$
   • B$4$
   • C$3$
   • D$6$

   (c)

   Show that the set $\{1,2,3,4,5\}$ under multiplication modulo 6 is not a group.

   [2 marks]

   Total for question 2: 4 marks

3. 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.

   (a)

   Find $rs$ and $sr$, writing each as a permutation, and state what this shows about the group.

   [3 marks]

   (b)

   Find $r^{-1}$ and show that $srs=r^{-1}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $G=\{0,1,2,3,4,5\}$ under addition modulo 6.

   (a)

   Show that $G$ satisfies the four group axioms.

   [6 marks]

   (b)

   Show that $G$ is cyclic by finding two different generators, and show that $2$ is not a generator.

   [6 marks]

   Total for question 4: 12 marks

End of questions