DP Mathematics HL Questionbank

• Syllabus

8.10

Sub sections and their related questions

Permutations under composition of permutations.

• SPNone.3srg.hl.TZ0.3b: The group $\{ K,{\text{ }} \circ \}$ is defined on the six permutations of the integers 1, 2,...
• 10M.3srg.hl.TZ0.4: The permutation ${p_1}$ of the set {1, 2, 3, 4} is defined...
• 14N.3srg.hl.TZ0.3a: Two members of $A$ are given by $p = (1{\text{ }}2{\text{ }}5)$ and...
• 14N.3srg.hl.TZ0.3c: Let $P =$ {all permutations in $A$ where exactly two integers change position}, and...
• 15N.3srg.hl.TZ0.3c: Find (i)     $p \circ p$; (ii)     the inverse of $p \circ p$.
• 16N.3srg.hl.TZ0.1a: (i)     Write the permutation...
• 16N.3srg.hl.TZ0.1b: (i)     Write the permutation...
• 16N.3srg.hl.TZ0.1c: Write the permutation $\alpha  \circ \beta$ as a composition of disjoint cycles.
• 16N.3srg.hl.TZ0.1d: Write the permutation $\beta  \circ \alpha$ as a composition of disjoint cycles.
• 16N.3srg.hl.TZ0.1e: State the order of $\{ G,{\text{ }} \circ \}$.
• 16N.3srg.hl.TZ0.1f: Find the number of permutations in $\{ G,{\text{ }} \circ \}$ which will result in $A$,...
• 18M.3srg.hl.TZ0.4a: Determine the order of S4.
• 18M.3srg.hl.TZ0.4b: Find the proper subgroup H of order 6 containing ${p_1}$, ${p_2}$ and their compositions....

Cycle notation for permutations.

• 14N.3srg.hl.TZ0.3a: Two members of $A$ are given by $p = (1{\text{ }}2{\text{ }}5)$ and...
• 15N.3srg.hl.TZ0.3a: Find the order of $\{ G,{\text{ }} \circ \}$.
• 15N.3srg.hl.TZ0.3b: State the identity element in $\{ G,{\text{ }} \circ \}$.
• 15N.3srg.hl.TZ0.3d: (i)     Find the maximum possible order of an element in $\{ H,{\text{ }} \circ \}$. (ii)    ...
• 16N.3srg.hl.TZ0.1a: (i)     Write the permutation...
• 16N.3srg.hl.TZ0.1b: (i)     Write the permutation...
• 16N.3srg.hl.TZ0.1c: Write the permutation $\alpha  \circ \beta$ as a composition of disjoint cycles.
• 16N.3srg.hl.TZ0.1d: Write the permutation $\beta  \circ \alpha$ as a composition of disjoint cycles.
• 16N.3srg.hl.TZ0.1e: State the order of $\{ G,{\text{ }} \circ \}$.
• 16N.3srg.hl.TZ0.1f: Find the number of permutations in $\{ G,{\text{ }} \circ \}$ which will result in $A$,...
• 18M.3srg.hl.TZ0.4a: Determine the order of S4.
• 18M.3srg.hl.TZ0.4b: Find the proper subgroup H of order 6 containing ${p_1}$, ${p_2}$ and their compositions....

Result that every permutation can be written as a composition of disjoint cycles.

• 15N.3srg.hl.TZ0.3d: (i)     Find the maximum possible order of an element in $\{ H,{\text{ }} \circ \}$. (ii)    ...
• 16N.3srg.hl.TZ0.1a: (i)     Write the permutation...
• 16N.3srg.hl.TZ0.1b: (i)     Write the permutation...
• 16N.3srg.hl.TZ0.1c: Write the permutation $\alpha  \circ \beta$ as a composition of disjoint cycles.
• 16N.3srg.hl.TZ0.1d: Write the permutation $\beta  \circ \alpha$ as a composition of disjoint cycles.
• 16N.3srg.hl.TZ0.1e: State the order of $\{ G,{\text{ }} \circ \}$.
• 16N.3srg.hl.TZ0.1f: Find the number of permutations in $\{ G,{\text{ }} \circ \}$ which will result in $A$,...
• 18M.3srg.hl.TZ0.4a: Determine the order of S4.
• 18M.3srg.hl.TZ0.4b: Find the proper subgroup H of order 6 containing ${p_1}$, ${p_2}$ and their compositions....

The order of a combination of cycles.

• 14N.3srg.hl.TZ0.3b: State a permutation belonging to $A$ of order (i)     $4$; (ii)     $6$.
• 15N.3srg.hl.TZ0.3a: Find the order of $\{ G,{\text{ }} \circ \}$.
• 15N.3srg.hl.TZ0.3d: (i)     Find the maximum possible order of an element in $\{ H,{\text{ }} \circ \}$. (ii)    ...
• 18M.3srg.hl.TZ0.4a: Determine the order of S4.
• 18M.3srg.hl.TZ0.4b: Find the proper subgroup H of order 6 containing ${p_1}$, ${p_2}$ and their compositions....