DP Mathematics HL Questionbank

• Syllabus

8.12

Sub sections and their related questions

Definition of a group homomorphism.

• SPNone.3srg.hl.TZ0.4: The groups \(\{ K,{\text{ }} * \} \) and \(\{ H,{\text{ }} \odot \} \) are defined by the...
• 14M.3srg.hl.TZ0.4b: (i)     Prove that the kernel of \(f,{\text{ }}K = {\text{Ker}}(f)\), is closed under the group...
• 14M.3srg.hl.TZ0.4a: Prove that \(f({e_G}) = {e_H}\), where \({e_G}\) is the identity element in \(G\) and \({e_H}\)...
• 14N.3srg.hl.TZ0.4a: Prove that for all \(a \in G,{\text{ }}f({a^{ - 1}}) = {\left( {f(a)} \right)^{ - 1}}\).
• 14N.3srg.hl.TZ0.4b: Let \(\{ H,{\text{ }} \circ \} \) be the cyclic group of order seven, and let \(p\) be a...
• 14N.3srg.hl.TZ0.4c: Given that \(f(x * y) = p\), find \(f(y)\).
• 15N.3srg.hl.TZ0.5a: Prove that the function \(f\) is a homomorphism from the group...
• 16M.3srg.hl.TZ0.3: The group \(\{ G,{\text{ }} * \} \) is Abelian and the bijection \(f:{\text{ }}G \to G\) is...
• 18M.3srg.hl.TZ0.4c: Let \(f{\text{:}}\,{S_4} \to {S_4}\) be defined by \(f\left( p \right) = p \circ p\)...

Definition of the kernel of a homomorphism.

• 14M.3srg.hl.TZ0.4b: (i)     Prove that the kernel of \(f,{\text{ }}K = {\text{Ker}}(f)\), is closed under the group...
• 14M.3srg.hl.TZ0.4a: Prove that \(f({e_G}) = {e_H}\), where \({e_G}\) is the identity element in \(G\) and \({e_H}\)...
• 15N.3srg.hl.TZ0.5b: Find the kernel of \(f\).
• 16M.3srg.hl.TZ0.3: The group \(\{ G,{\text{ }} * \} \) is Abelian and the bijection \(f:{\text{ }}G \to G\) is...
• 18M.3srg.hl.TZ0.4c: Let \(f{\text{:}}\,{S_4} \to {S_4}\) be defined by \(f\left( p \right) = p \circ p\)...

Proof that the kernel and range of a homomorphism are subgroups.

• 15N.3srg.hl.TZ0.5c: Prove that \(\{ {\text{Ker}}(f),{\text{ }} + \} \) is a subgroup of...
• 16M.3srg.hl.TZ0.3: The group \(\{ G,{\text{ }} * \} \) is Abelian and the bijection \(f:{\text{ }}G \to G\) is...
• 18M.3srg.hl.TZ0.4c: Let \(f{\text{:}}\,{S_4} \to {S_4}\) be defined by \(f\left( p \right) = p \circ p\)...

Proof of homomorphism properties for identities and inverses.

• 16M.3srg.hl.TZ0.3: The group \(\{ G,{\text{ }} * \} \) is Abelian and the bijection \(f:{\text{ }}G \to G\) is...
• 18M.3srg.hl.TZ0.4c: Let \(f{\text{:}}\,{S_4} \to {S_4}\) be defined by \(f\left( p \right) = p \circ p\)...

Isomorphism of groups.

• 12N.3srg.hl.TZ0.3d: Is \(\{ P(A){\text{, }}\Delta \} \) isomorphic to \(\{ {\mathbb{Z}_4},{\text{ }}{ + _4}\} \) ?...
• 08M.3srg.hl.TZ1.3: (a)     Find the six roots of the equation \({z^6} - 1 = 0\) , giving your answers in the form...
• 11M.3srg.hl.TZ0.5b: Consider the group G , of order 4, which has distinct elements a , b and c and the identity...
• SPNone.3srg.hl.TZ0.3b: The group \(\{ K,{\text{ }} \circ \} \) is defined on the six permutations of the integers 1, 2,...
• 10M.3srg.hl.TZ0.3: (a)     Consider the set A = {1, 3, 5, 7} under the binary operation \( * \), where \( * \)...
• 10N.3srg.hl.TZ0.4: Set...
• 11N.3srg.hl.TZ0.1c: Show that \(\{ G,{\text{ }}{ \times _{16}}\} \) and \(\{ H,{\text{ }} * \} \) are not isomorphic.
• 15M.3srg.hl.TZ0.5c: The mapping \(\phi :G \to G\) is given by \(\phi (g) = g + g\), for \(g \in G\). Prove that...
• 16M.3srg.hl.TZ0.3: The group \(\{ G,{\text{ }} * \} \) is Abelian and the bijection \(f:{\text{ }}G \to G\) is...
• 18M.3srg.hl.TZ0.4c: Let \(f{\text{:}}\,{S_4} \to {S_4}\) be defined by \(f\left( p \right) = p \circ p\)...

The order of an element is unchanged by an isomorphism.

• 16M.3srg.hl.TZ0.3: The group \(\{ G,{\text{ }} * \} \) is Abelian and the bijection \(f:{\text{ }}G \to G\) is...
• 18M.3srg.hl.TZ0.4c: Let \(f{\text{:}}\,{S_4} \to {S_4}\) be defined by \(f\left( p \right) = p \circ p\)...