Discrete Mathematics 7: Binary operations and groupsAQA A-Level Further Maths: Topic test

1. 1
   The binary operation $\star$ is defined on the set $S=\{0,1,2,3,4\}$ by $x\star y=$ the remainder when $x+y+xy$ is divided by 5.

   (a)

   What is the value of $3\star4$?

   [1 mark]

   • A2
   • B3
   • C4
   • D0

   (b)

   Which element of $S$ has no inverse under $\star$? (The identity is 0.)

   [1 mark]

   • A1
   • B2
   • C3
   • D4

   (c)

   Write down the row of the Cayley table for $\star$ that gives $2\star y$ for $y=0,1,2,3,4$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The set $G=\{1,3,7,9\}$ is a group under multiplication modulo 10, written $\otimes$.

   (a)

   What is the inverse of 3 in $G$?

   [1 mark]

   • A7
   • B9
   • C3
   • D1

   (b)

   How many elements of $G$ are their own inverse?

   [1 mark]

   • A1
   • B2
   • C3
   • D4

   (c)

   Use the inverse of 3 to solve $3\otimes x=9$ for $x$ in $G$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $A=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$, which represents an anticlockwise rotation through $90^\circ$ about the origin. The set $G=\{I,A,A^2,A^3\}$ is under matrix multiplication, where $I$ is the identity matrix and $A^4=I$.

   (a)

   Show that $G$ satisfies the four group axioms.

   [3 marks]

   (b)

   Find all the subgroups of $G$, and state which of them are proper and non-trivial.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The set $G=\{1,2,4,5,7,8\}$ is a group under multiplication modulo 9.

   (a)

   Identify the identity element, find the inverse of each element, and state the order of each element.

   [6 marks]

   (b)

   Use Lagrange's theorem and the orders of the elements to show that $G$ is cyclic, state its generators and find all of its subgroups.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The set $G=\{1,2,4,5,8,10,11,13,16,17,19,20\}$ is a group of order 12 under multiplication modulo 21.

   (a)

   What is the order of the element 2?

   [1 mark]

   • A3
   • B4
   • C6
   • D12

   (b)

   Which statement about $G$ is correct?

   [1 mark]

   • A$G$ is cyclic and generated by 2
   • B$G$ is cyclic and generated by 5
   • C$G$ is cyclic because its order is 12
   • D$G$ is not cyclic because no element has order 12

   (c)

   Write down the elements of the subgroup generated by 4 and state, using Lagrange's theorem, how many left cosets it has in $G$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The group $R$ is the set of symmetries of a rectangle that is not a square: the identity $e$, the half turn $r$, the reflection $h$ in the horizontal axis and the reflection $v$ in the vertical axis, under composition. The group $M$ is the set $\{1,2,3,4\}$ under multiplication modulo 5.

   (a)

   How many elements of $R$ have order 2?

   [1 mark]

   • A1
   • B3
   • C2
   • D4

   (b)

   Which statement about $R$ and $M$ is correct?

   [1 mark]

   • AThey are not isomorphic because $M$ has an element of order 4 and $R$ does not
   • BThey are isomorphic because they both have four elements
   • CThey are not isomorphic because $R$ is not abelian
   • DThey are isomorphic because they are both abelian

   (c)

   Show that $M$ is cyclic by finding a generator.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The binary operation $\ast$ is defined on the set $T=\{0,1,2,3,4,5\}$ by $x\ast y=$ the remainder when $x+y+2$ is divided by 6.

   (a)

   Find the identity element of $T$ under $\ast$ and prove that it is the identity.

   [3 marks]

   (b)

   Prove that $\ast$ is associative on $T$, and find the inverse of 3.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The set $K=\{1,3,5,7,9,11,13,15\}$ is a group of order 8 under multiplication modulo 16, and the set $Q=\{1,3,7,9,11,13,17,19\}$ is a group of order 8 under multiplication modulo 20.

   (a)

   Find the order of every element of $K$. Hence show that $K$ is not cyclic, state the possible orders of its subgroups, and give one cyclic and one non-cyclic subgroup of order 4.

   [6 marks]

   (b)

   (i) Show that $K$ and $Q$ have the same number of elements of each order. (ii) A function $\phi:K\to Q$ sends $1,3,5,7,9,11,13,15$ to $1,3,13,11,9,7,17,19$ respectively. Verify that $\phi(5\times11)=\phi(5)\times\phi(11)$.

   [6 marks]

   Total for question 8: 12 marks

End of questions