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 $f\left( p \right) = p \circ p$...