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....