DP Mathematics HL Questionbank

• Syllabus

8.7

Sub sections and their related questions

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

• 12M.3srg.hl.TZ0.1a: Associativity and commutativity are two of the five conditions for a set S with the binary...
• 12M.3srg.hl.TZ0.1b: The Cayley table for the binary operation \( \odot \) defined on the set T = {p, q, r, s, t} is...
• 12N.3srg.hl.TZ0.3c: Prove that \(\left\{ {P(A),{\text{ }}\Delta } \right\}\) is a group. You are allowed to assume...
• 12N.3srg.hl.TZ0.3e: (i)     State the identity element for \(\{ P(S){\text{, }}\Delta \} \). (ii)     Write down...
• 12N.3srg.hl.TZ0.3f: Explain why \(\{ P(S){\text{, }} \cup \} \) is not a group.
• 12N.3srg.hl.TZ0.3g: Explain why \(\{ P(S){\text{, }} \cap \} \) is not a group.
• 12N.3srg.hl.TZ0.4h: Explain why \(\{ G, * \} \) is an Abelian group.
• 08M.3srg.hl.TZ2.1: (a)     Draw the Cayley table for the set of integers G = {0, 1, 2, 3, 4, 5} under addition...
• 11M.3srg.hl.TZ0.1b: (i)     Show that {S , \( * \)} is a group. (ii)     Find the order of each element of {S ,...
• 10M.3srg.hl.TZ0.3: (a)     Consider the set A = {1, 3, 5, 7} under the binary operation \( * \), where \( * \)...
• 13M.3srg.hl.TZ0.2b: Give one reason why \(\{ S,{\text{ }}{ \times _{14}}\} \) is not a group.
• 13M.3srg.hl.TZ0.2c: Show that a new set G can be formed by removing one of the elements of S such that...
• 11N.3srg.hl.TZ0.1a: Consider the following Cayley table for the set G = {1, 3, 5, 7, 9, 11, 13, 15} under the...
• 11N.3srg.hl.TZ0.1b: The Cayley table for the set...
• 11N.3srg.hl.TZ0.4b: The Cayley table for G is shown below.     The subgroup H is given as...
• 14M.3srg.hl.TZ0.2a: (i)     Write down the six smallest non-negative elements of \(S\). (ii)     Show that...
• 15M.3srg.hl.TZ0.5a: Show that \((G,{\text{ }} + )\) forms a group where \( + \) denotes addition on \(\mathbb{Q}\)....
• 15N.3srg.hl.TZ0.4b: Prove that \(\{ T,{\text{ }} * \} \) forms an Abelian group.
• 16M.3srg.hl.TZ0.1b: Show that \(\{ S,{\text{ }} * \} \) is an Abelian group.
• 17M.3srg.hl.TZ0.4a: Show that \(\{ \mathbb{Z},{\text{ }} * \} \) is an Abelian group.
• 18M.3srg.hl.TZ0.1a: Show that {T, ×10} is a group. (You may assume associativity.)
• 18M.3srg.hl.TZ0.1b: By making reference to the Cayley table, explain why T is Abelian.

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

• 12M.3srg.hl.TZ0.1a: Associativity and commutativity are two of the five conditions for a set S with the binary...
• 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.TZ2.5: (a)     Write down why the table below is a Latin...
• 16M.3srg.hl.TZ0.1b: Show that \(\{ S,{\text{ }} * \} \) is an Abelian group.
• 17M.3srg.hl.TZ0.4a: Show that \(\{ \mathbb{Z},{\text{ }} * \} \) is an Abelian group.
• 18M.3srg.hl.TZ0.1a: Show that {T, ×10} is a group. (You may assume associativity.)
• 18M.3srg.hl.TZ0.1b: By making reference to the Cayley table, explain why T is Abelian.

Abelian groups.

• 12M.3srg.hl.TZ0.1a: Associativity and commutativity are two of the five conditions for a set S with the binary...
• 12M.3srg.hl.TZ0.1b: The Cayley table for the binary operation \( \odot \) defined on the set T = {p, q, r, s, t} is...
• 09M.3srg.hl.TZ0.1: (a)     Show that {1, −1, i, −i} forms a group of complex numbers G under multiplication. (b)  ...
• 10N.3srg.hl.TZ0.4: Set...
• 15N.3srg.hl.TZ0.4b: Prove that \(\{ T,{\text{ }} * \} \) forms an Abelian group.
• 16M.3srg.hl.TZ0.1b: Show that \(\{ S,{\text{ }} * \} \) is an Abelian group.
• 17M.3srg.hl.TZ0.4a: Show that \(\{ \mathbb{Z},{\text{ }} * \} \) is an Abelian group.
• 18M.3srg.hl.TZ0.1a: Show that {T, ×10} is a group. (You may assume associativity.)
• 18M.3srg.hl.TZ0.1b: By making reference to the Cayley table, explain why T is Abelian.