DP Further Mathematics HL Questionbank

• Syllabus

4.7

Sub sections and their related questions

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

• 11M.2.hl.TZ0.6a: (i)     Draw the Cayley table for the set \(S = \left\{ {0,1,2,3,4,\left. 5 \right\}} \right.\)...
• 10M.2.hl.TZ0.1b: The domain of \( * \) is now reduced to \(S = \left\{ {0,2,3,4,5,\left. 6 \right\}} \right.\) and...
• 09M.2.hl.TZ0.2A.a: (i)     Show that \({\mathbb{Z}_4}\) (the set of integers modulo 4) together with the operation...
• 09M.2.hl.TZ0.2B.b: the group is cyclic.
• 13M.1.hl.TZ0.2a: Show that \({(ba)^2} = e\) .
• 13M.1.hl.TZ0.2b: Express \({(bab)^{ - 1}}\) in its simplest form.
• 13M.1.hl.TZ0.2c: Given that \(a \ne e\) ,   (i)     show that \(b \ne e\) ;   (ii)     show that \(G\) is not...
• 13M.2.hl.TZ0.5e: Determine whether the set \(E\) forms a group under   (i)     the operation of addition;   (ii)...
• 07M.1.hl.TZ0.3a: Show that the set \(S\) of numbers of the form \({2^m} \times {3^n}\) , where...
• 07M.2.hl.TZ0.5b: Let \(S\) denote the set...
• 12M.1.hl.TZ0.1a: The set \({{\rm{S}}_1} = \left\{ {2,4,6,8} \right\}\) and \({ \times _{10}}\) denotes...
• SPNone.1.hl.TZ0.14b: Show that \(S\) forms an Abelian group under matrix multiplication.
• SPNone.2.hl.TZ0.4b: Show that \(\left\{ {S,{ \times _9}} \right\}\) is not a group.
• SPNone.2.hl.TZ0.4c: Prove that a group \(\left\{ {G,{ \times _9}} \right\}\) can be formed by removing two elements...
• SPNone.2.hl.TZ0.4e: Solve the equation \(4{ \times _9}x{ \times _9}x = 1\) .
• 14M.2.hl.TZ0.2a: The set \(S\) contains the eighth roots of unity given by...
• 15M.1.hl.TZ0.3b: Show that \(\{ S,{\text{ }}{ + _6}\} \) forms an Abelian group.
• 18M.1.hl.TZ0.4a: Copy and complete the following Cayley table for the transformations of T1, T2, T3, T4, under the...
• 18M.1.hl.TZ0.4b.i: Show that T1, T2, T3, T4 under the operation of composition of transformations form a group....
• 18M.1.hl.TZ0.4b.ii: Show that this group is cyclic.

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

• 18M.1.hl.TZ0.4a: Copy and complete the following Cayley table for the transformations of T1, T2, T3, T4, under the...
• 18M.1.hl.TZ0.4b.i: Show that T1, T2, T3, T4 under the operation of composition of transformations form a group....
• 18M.1.hl.TZ0.4b.ii: Show that this group is cyclic.

Abelian groups.

• 08M.1.hl.TZ0.2c: Solve the equation \(x * 6 * x = 3\) where \(x \in G\) .
• 15M.1.hl.TZ0.3b: Show that \(\{ S,{\text{ }}{ + _6}\} \) forms an Abelian group.
• 15M.2.hl.TZ0.9c: Show that if \(G\) is Abelian, then \(H\) must also be Abelian.
• 18M.1.hl.TZ0.4a: Copy and complete the following Cayley table for the transformations of T1, T2, T3, T4, under the...
• 18M.1.hl.TZ0.4b.i: Show that T1, T2, T3, T4 under the operation of composition of transformations form a group....
• 18M.1.hl.TZ0.4b.ii: Show that this group is cyclic.