Group Homomorphism Quiz April 22, 2025April 22, 2025 cracksadmin Welcome to your Group Homomorphism Quiz Let $( \phi: G \to H )$ be a group homomorphism. If $( G )$ is abelian, which of the following must be true about $( H )?$ $( H ) $must be abelian. $( H )$ must be cyclic. The image $( \phi(G) )$ must be abelian. The kernel $( \ker(\phi) )$ must be cyclic. None Let $( \phi: G \to H )$ be an injective group homomorphism. Which of the following is always true? $( \phi )$ is an isomorphism. $( \ker(\phi) = G ).$ $( \ker(\phi) = {e_G} ).$ $( \phi(G) = H ).$ None Let $( \phi: \mathbb{Z} \to \mathbb{Z}/n\mathbb{Z} )$ be defined by $( \phi(k) = k \mod n ).$ What is the kernel of $( \phi )?$ $( {0} )$ $C$ $( n\mathbb{Z} )$ $( \mathbb{Z}/n\mathbb{Z} )$ None If $( \phi: G \to H )$ is a group homomorphism and $( N )$ is a normal subgroup of $( G )$, what can be said about $( \phi(N) )?$ $( \phi(N) )$ is always a normal subgroup of $( H ).$ $( \phi(N) )$ is a normal subgroup of $( \phi(G) ).$ $( \phi(N) )$ is a subgroup of $( H ) $ but not necessarily normal. $( \phi(N) )$is always the trivial subgroup. None Consider the homomorphism $( \phi: \mathbb{Z} \to \mathbb{Z} )$ defined by $( \phi(n) = 2n ).$ What is the quotient group $( \mathbb{Z}/\ker(\phi) )?$ $( \mathbb{Z} )$ $( \mathbb{Z}/2\mathbb{Z} )$ $( {0} )$ $( \mathbb{Z}/4\mathbb{Z} )$ None Consider the homomorphism $( \phi: \mathbb{Z}/6\mathbb{Z} \to \mathbb{Z}/3\mathbb{Z} )$ defined by $( \phi([a]_6) = [a]_3 ).$ What is the image of $( \phi )?$ $( {[0]_3} )$ $( {[0]_3, [1]_3} )$ $( \mathbb{Z}/3\mathbb{Z} )$ $( {[1]_3, [2]_3} )$ None Let $( \phi: G \to H )$ be a homomorphism, and let $( K = \ker(\phi) )$. If $( G/K \cong H ),$ what can be said about $( \phi )?$ $( \phi )$ is injective. $( \phi )$ is surjective. $( \phi )$ is bijective. $( \phi )$ is the zero map. None Let $( \phi: \mathbb{R} \to \mathbb{R} )$ be a group homomorphism under addition. Which of the following is true about $( \phi )?$ $( \phi )$ must be the zero map. $( \phi )$ must be an isomorphism. $( \phi(x) = cx )$ for some $( c \in \mathbb{R} ).$ $( \phi )$ must be surjective. None If $( \phi: G \to H )$ is a surjective homomorphism and $( G )$ is finite, what can be said about $( |G| )$ and $( |H| )?$ $( |G| = |H| )$ $( |G| ) divides ( |H| )$ $( |H| ) divides ( |G| )$ $( |G| \leq |H| )$ None Let $( \phi: S_3 \to \mathbb{Z}/2\mathbb{Z} )$ be a group homomorphism, where $( S_3 )$ is the symmetric group on $3$ elements. What is the possible order of $( \ker(\phi) )?$ 1 or 3 1 or 6 1, 2, or 3 1, 3, or 6 None 1 out of 2 Time's upTime is Up!
