DP Further Mathematics HL Questionbank

• Syllabus

4.7

Sub sections and their related questions

The definition of a group $\left\{ {G, * } \right\}$ .

• 11M.2.hl.TZ0.6a: (i)     Draw the Cayley table for the set $S = \left\{ {0,1,2,3,4,\left. 5 \right\}} \right.$...
• 10M.2.hl.TZ0.1b: The domain of $*$ is now reduced to $S = \left\{ {0,2,3,4,5,\left. 6 \right\}} \right.$ and...
• 09M.2.hl.TZ0.2A.a: (i)     Show that ${\mathbb{Z}_4}$ (the set of integers modulo 4) together with the operation...
• 09M.2.hl.TZ0.2B.b: the group is cyclic.
• 13M.1.hl.TZ0.2a: Show that ${(ba)^2} = e$ .
• 13M.1.hl.TZ0.2b: Express ${(bab)^{ - 1}}$ in its simplest form.
• 13M.1.hl.TZ0.2c: Given that $a \ne e$ ,   (i)     show that $b \ne e$ ;   (ii)     show that $G$ is not...
• 13M.2.hl.TZ0.5e: Determine whether the set $E$ forms a group under   (i)     the operation of addition;   (ii)...
• 07M.1.hl.TZ0.3a: Show that the set $S$ of numbers of the form ${2^m} \times {3^n}$ , where...
• 07M.2.hl.TZ0.5b: Let $S$ denote the set...
• 12M.1.hl.TZ0.1a: The set ${{\rm{S}}_1} = \left\{ {2,4,6,8} \right\}$ and ${ \times _{10}}$ denotes...
• SPNone.1.hl.TZ0.14b: Show that $S$ forms an Abelian group under matrix multiplication.
• SPNone.2.hl.TZ0.4b: Show that $\left\{ {S,{ \times _9}} \right\}$ is not a group.
• SPNone.2.hl.TZ0.4c: Prove that a group $\left\{ {G,{ \times _9}} \right\}$ can be formed by removing two elements...
• SPNone.2.hl.TZ0.4e: Solve the equation $4{ \times _9}x{ \times _9}x = 1$ .
• 14M.2.hl.TZ0.2a: The set $S$ contains the eighth roots of unity given by...
• 15M.1.hl.TZ0.3b: Show that $\{ S,{\text{ }}{ + _6}\}$ forms an Abelian group.
• 18M.1.hl.TZ0.4a: Copy and complete the following Cayley table for the transformations of T1, T2, T3, T4, under the...
• 18M.1.hl.TZ0.4b.i: Show that T1, T2, T3, T4 under the operation of composition of transformations form a group....
• 18M.1.hl.TZ0.4b.ii: Show that this group is cyclic.

The operation table of a group is a Latin square, but the converse is false.

• 18M.1.hl.TZ0.4a: Copy and complete the following Cayley table for the transformations of T1, T2, T3, T4, under the...
• 18M.1.hl.TZ0.4b.i: Show that T1, T2, T3, T4 under the operation of composition of transformations form a group....
• 18M.1.hl.TZ0.4b.ii: Show that this group is cyclic.

Abelian groups.

• 08M.1.hl.TZ0.2c: Solve the equation $x * 6 * x = 3$ where $x \in G$ .
• 15M.1.hl.TZ0.3b: Show that $\{ S,{\text{ }}{ + _6}\}$ forms an Abelian group.
• 15M.2.hl.TZ0.9c: Show that if $G$ is Abelian, then $H$ must also be Abelian.
• 18M.1.hl.TZ0.4a: Copy and complete the following Cayley table for the transformations of T1, T2, T3, T4, under the...
• 18M.1.hl.TZ0.4b.i: Show that T1, T2, T3, T4 under the operation of composition of transformations form a group....
• 18M.1.hl.TZ0.4b.ii: Show that this group is cyclic.