DP Mathematics HL Questionbank

• Syllabus

8.11

Sub sections and their related questions

Subgroups, proper subgroups.

• 12M.3srg.hl.TZ0.1b: The Cayley table for the binary operation \( \odot \) defined on the set T = {p, q, r, s, t} is...
• 08M.3srg.hl.TZ2.1: (a)     Draw the Cayley table for the set of integers G = {0, 1, 2, 3, 4, 5} under addition...
• 08M.3srg.hl.TZ2.4b: Given the group \((G,{\text{ }} * )\), a subgroup \((H,{\text{ }} * )\) and...
• 11M.3srg.hl.TZ0.1c: The set T is defined by \(\{ x * x:x \in S\} \). Show that {T , \( * \)} is a subgroup 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)    ...
• SPNone.3srg.hl.TZ0.5: Let \(\{ G,{\text{ }} * \} \) be a finite group and let H be a non-empty subset of G . Prove that...
• 10N.3srg.hl.TZ0.4: Set...
• 10N.3srg.hl.TZ0.5: Let \(\{ G,{\text{ }} * \} \) be a finite group that contains an element a (that is not the...
• 13M.3srg.hl.TZ0.2e: Find the proper subgroups of \(\{ G,{\text{ }}{ \times _{14}}\} \).
• 13M.3srg.hl.TZ0.5: H and K are subgroups of a group G. By considering the four group axioms, prove that \(H \cap K\)...
• 11N.3srg.hl.TZ0.1b: The Cayley table for the set...
• 13N.3srg.hl.TZ0.2b: State, with a reason, whether or not it is necessary that a group is cyclic given that all its...
• 14N.3srg.hl.TZ0.1c: Write down the three sets that form subgroups of order 2.
• 14N.3srg.hl.TZ0.1d: Find the three sets that form subgroups of order 4.
• 15M.3srg.hl.TZ0.1c: Write down the subgroups containing (i)     \(r\), (ii)     \(u\).
• 15M.3srg.hl.TZ0.5b: Assuming that \((H,{\text{ }} + )\) forms a group, show that it is a proper subgroup of...
• 15N.3srg.hl.TZ0.3a: Find the order of \(\{ G,{\text{ }} \circ \} \).
• 16M.3srg.hl.TZ0.1d: (i)     Find the two proper subgroups of \(\{ S,{\text{ }} * \} \). (ii)     Find the coset of...
• 16M.3srg.hl.TZ0.1e: Solve the equation \(2 * x * 4 * x * 4 = 2\).
• 16M.3srg.hl.TZ0.5: The group \(\{ G,{\text{ }} * \} \) is defined on the set \(G\) with binary operation \( * \)....
• 17N.3srg.hl.TZ0.1b: Write down the elements in set \(K\).

Use and proof of subgroup tests.

• 15N.3srg.hl.TZ0.5c: Prove that \(\{ {\text{Ker}}(f),{\text{ }} + \} \) is a subgroup of...
• 16M.3srg.hl.TZ0.1d: (i)     Find the two proper subgroups of \(\{ S,{\text{ }} * \} \). (ii)     Find the coset of...
• 16M.3srg.hl.TZ0.1e: Solve the equation \(2 * x * 4 * x * 4 = 2\).
• 16M.3srg.hl.TZ0.5: The group \(\{ G,{\text{ }} * \} \) is defined on the set \(G\) with binary operation \( * \)....
• 17N.3srg.hl.TZ0.1b: Write down the elements in set \(K\).

Definition and examples of left and right cosets of a subgroup of a group.

• 14M.3srg.hl.TZ0.4c: (i)     Prove that \(gk{g^{ - 1}} \in K\) for all \(g \in G,{\text{ }}k \in K\). (ii)     Deduce...
• 14N.3srg.hl.TZ0.5d: Let \(\{ H,{\rm{ }} * {\rm{\} }}\) be a subgroup of \(\{ G,{\rm{ }} * {\rm{\} }}\) .Let \(R\) be...
• 14N.3srg.hl.TZ0.5e: Let \(\{ H,{\rm{ }} * {\rm{\} }}\) be a subgroup of \(\{ G,{\rm{ }} * {\rm{\} }}\) .Let \(R\) be...
• 16M.3srg.hl.TZ0.1d: (i)     Find the two proper subgroups of \(\{ S,{\text{ }} * \} \). (ii)     Find the coset of...
• 16M.3srg.hl.TZ0.1e: Solve the equation \(2 * x * 4 * x * 4 = 2\).
• 16M.3srg.hl.TZ0.5: The group \(\{ G,{\text{ }} * \} \) is defined on the set \(G\) with binary operation \( * \)....
• 17N.3srg.hl.TZ0.1b: Write down the elements in set \(K\).

Lagrange’s theorem.

• 08M.3srg.hl.TZ2.5: (a)     Write down why the table below is a Latin...
• 14N.3srg.hl.TZ0.5a: State Lagrange’s theorem.
• 14N.3srg.hl.TZ0.5e: Let \(\{ H,{\rm{ }} * {\rm{\} }}\) be a subgroup of \(\{ G,{\rm{ }} * {\rm{\} }}\) .Let \(R\) be...
• 16M.3srg.hl.TZ0.1d: (i)     Find the two proper subgroups of \(\{ S,{\text{ }} * \} \). (ii)     Find the coset of...
• 16M.3srg.hl.TZ0.1e: Solve the equation \(2 * x * 4 * x * 4 = 2\).
• 16M.3srg.hl.TZ0.5: The group \(\{ G,{\text{ }} * \} \) is defined on the set \(G\) with binary operation \( * \)....
• 17N.3srg.hl.TZ0.1b: Write down the elements in set \(K\).

Use and proof of the result that the order of a finite group is divisible by the order of any element. (Corollary to Lagrange’s theorem.)

• 15M.3srg.hl.TZ0.2b: (i)     Find the least solution of \(x * x = e\). (ii)     Deduce that \((S,{\text{ }} * )\) is...
• 17N.3srg.hl.TZ0.1b: Write down the elements in set \(K\).