DP Mathematics HL Questionbank

• Syllabus

8.9

Sub sections and their related questions

The order of a group.

• SPNone.3srg.hl.TZ0.3a: (i)     Write down the Cayley table for $\{ G,{\text{ }}{ \times _7}\}$ . (ii)     Determine...
• 13M.3srg.hl.TZ0.2d: Determine the order of each element of $\{ G,{\text{ }}{ \times _{14}}\}$.
• 11N.3srg.hl.TZ0.1d: Show that $\{ H,{\text{ }} * \}$ is not cyclic.
• 13N.3srg.hl.TZ0.2a: (i)      Prove that $G$ is cyclic and state two of its generators. (ii)     Let $H$ be the...
• 15N.3srg.hl.TZ0.3a: Find the order of $\{ G,{\text{ }} \circ \}$.
• 16M.3srg.hl.TZ0.1c: Determine the orders of all the elements of $\{ S,{\text{ }} * \}$.
• 16N.3srg.hl.TZ0.3a: State the possible orders of an element of $\{ G,{\text{ }} * \}$ and for each order give an...
• 16N.3srg.hl.TZ0.3b: (i)     State a generator for $\{ H,{\text{ }} * \}$. (ii)     Write down the elements of...
• 18M.3srg.hl.TZ0.1c.i: Find the order of each element of {T, ×10}.
• 18M.3srg.hl.TZ0.1c.ii: Hence show that {T, ×10} is cyclic and write down all its generators.
• 18M.3srg.hl.TZ0.1d: The binary operation multiplication modulo 10, denoted by ×10 , is defined on the set V = {1, 3...

The order of a group element.

• 12M.3srg.hl.TZ0.5a: (i)     Show that $gh{g^{ - 1}}$ has order 2 for all $g \in G$. (ii)     Deduce that gh = hg...
• 08M.3srg.hl.TZ1.5: Let $p = {2^k} + 1,{\text{ }}k \in {\mathbb{Z}^ + }$ be a prime number and let G be the group...
• 11M.3srg.hl.TZ0.1b: (i)     Show that {S , $*$} is a group. (ii)     Find the order of each element of {S ,...
• 09N.3srg.hl.TZ0.5: Let {G , $*$} be a finite group of order n and let H be a non-empty subset of G . (a)    ...
• 14N.3srg.hl.TZ0.1b: Find the order of each of the elements of the group.
• 15M.3srg.hl.TZ0.1b: (i)     State the inverse of each element. (ii)     Determine the order of each element.
• 15N.3srg.hl.TZ0.3d: (i)     Find the maximum possible order of an element in $\{ H,{\text{ }} \circ \}$. (ii)    ...
• 15N.3srg.hl.TZ0.4c: Find the order of each element in $T$.
• 16M.3srg.hl.TZ0.1c: Determine the orders of all the elements of $\{ S,{\text{ }} * \}$.
• 16N.3srg.hl.TZ0.3a: State the possible orders of an element of $\{ G,{\text{ }} * \}$ and for each order give an...
• 16N.3srg.hl.TZ0.3b: (i)     State a generator for $\{ H,{\text{ }} * \}$. (ii)     Write down the elements of...
• 18M.3srg.hl.TZ0.1c.i: Find the order of each element of {T, ×10}.
• 18M.3srg.hl.TZ0.1c.ii: Hence show that {T, ×10} is cyclic and write down all its generators.
• 18M.3srg.hl.TZ0.1d: The binary operation multiplication modulo 10, denoted by ×10 , is defined on the set V = {1, 3...

Cyclic groups.

• 08M.3srg.hl.TZ1.3: (a)     Find the six roots of the equation ${z^6} - 1 = 0$ , giving your answers in the form...
• 08M.3srg.hl.TZ2.1: (a)     Draw the Cayley table for the set of integers G = {0, 1, 2, 3, 4, 5} under addition...
• 09M.3srg.hl.TZ0.1: (a)     Show that {1, −1, i, −i} forms a group of complex numbers G under multiplication. (b)  ...
• 10M.3srg.hl.TZ0.5: Let G be a finite cyclic group. (a)     Prove that G is Abelian. (b)     Given that a is a...
• 10N.3srg.hl.TZ0.4: Set...
• 14N.3srg.hl.TZ0.4b: Let $\{ H,{\text{ }} \circ \}$ be the cyclic group of order seven, and let $p$ be a...
• 16M.3srg.hl.TZ0.1c: Determine the orders of all the elements of $\{ S,{\text{ }} * \}$.
• 16N.3srg.hl.TZ0.3a: State the possible orders of an element of $\{ G,{\text{ }} * \}$ and for each order give an...
• 16N.3srg.hl.TZ0.3b: (i)     State a generator for $\{ H,{\text{ }} * \}$. (ii)     Write down the elements of...
• 18M.3srg.hl.TZ0.1c.i: Find the order of each element of {T, ×10}.
• 18M.3srg.hl.TZ0.1c.ii: Hence show that {T, ×10} is cyclic and write down all its generators.
• 18M.3srg.hl.TZ0.1d: The binary operation multiplication modulo 10, denoted by ×10 , is defined on the set V = {1, 3...

Generators.

• 09M.3srg.hl.TZ0.1: (a)     Show that {1, −1, i, −i} forms a group of complex numbers G under multiplication. (b)  ...
• 10M.3srg.hl.TZ0.5: Let G be a finite cyclic group. (a)     Prove that G is Abelian. (b)     Given that a is a...

Proof that all cyclic groups are Abelian.

• 10M.3srg.hl.TZ0.5: Let G be a finite cyclic group. (a)     Prove that G is Abelian. (b)     Given that a is a...