\[Let\,\left( {G,} \right)\,be\,a\,group\,with\,subgroup\,H.\,For\,a,\,b \in G,\,a\,is\,congruent\,to\,b\,\bmod ulo\,H,\,and\,written\,a \equiv b\bmod H\,iff\,a{b^{ - 1}} \in H.\]

• A) True
• B) False
• C)
• D)

{\text{Let}}\,{\text{a}}\,{\text{projection}}\,{\text{map}}\,f:{Z_4} \times {Z_2} \to {Z_4}\,{\text{given}}\,{\text{by}}\,f(x,y) = x\,{\text{is}}\,{\text{a}}\,{\text{homomorphism}}\,{\text{where}}\,\ker (f)\,{\text{is}}\,\{ 0\} \times {Z_2},\,{\text{then}}\,{\text{which}}\,{\text{of}}\,{\text{the}}\,{\text{following,}}\,{\text{the}}\,{\text{factor}}\,{\text{group}}\,{Z_4} \times {Z_2}/\{ 0\} \times {Z_2}\,{\text{is}}\,{\text{isomorphic}}\,{\text{to}}?

• A) {Z_4}
• B) {Z_2}
• C) \{ 0\}
• D) {Z_4} \times {Z_2}

{\text{Let}}\,N\,{\text{be}}\,{\text{a}}\,{\text{normal}}\,{\text{subgroup}}\,{\text{of}}\,G.\,{\text{In}}\,{\text{the}}\,{\text{factor}}\,{\text{group}}\,G/N,\,{\text{the}}\,{\text{subgroup}}\,N\,{\text{acts}}\,{\text{as}}\,{\text{identity}}\,{\text{element}}.

• A) {\text{False}}
• B) {\text{True}}
• C)
• D)

{\text{Let G be a group and H be a subgroup of G then H is normal iff }}\left( {{\text{aH}}} \right)\left( {{\text{bH}}} \right) = \left( {{\text{ab}}} \right){\text{H}},{\text{ for all a}},{\text{b}} \in {\text{G}}.

• A) True
• B) False
• C)
• D)

{\text{Let}}\,G\,{\text{be}}\,{\text{a}}\,{\text{group}}\,{\text{and}}\,N\,{\text{is}}\,{\text{a}}\,{\text{normal}}\,{\text{subgroup}}\,{\text{of}}\,G.\,{\text{Which}}\,{\text{statement}}\,{\text{is}}\,{\text{not}}\,{\text{true}}?

• A) G/N \cong G\,{\text{if}}\,N = \{ e\}
• B) G/G \cong \{ e\}
• C) G/\{ e\} \cong \{ e\}
• D) G/\{ e\} \cong G

\[{\text{Let G be a group and H be a subgroup of G then H is normal iff }}\left( {{\text{aH}}} \right)\left( {{\text{bH}}} \right) = \left( {{\text{ab}}} \right){\text{H}},{\text{ for all a}},{\text{b}} \in {\text{G}}.\]

• A) True
• B) False
• C)
• D)

\[{\text{A homomorphism }}\varphi :G \to G'\,is{\text{ injective iff }}\ker {\text{(}}\varphi ) = \left\{ e \right\}{\text{.}}\]

• A) False
• B) True
• C)
• D)

{\text{The}}\,{\text{trivial}}\,{\text{subgroup}}\,N = \{ 0\} \,{\text{of}}\,Z\,{\text{is}}\,{\text{not}}\,{\text{a}}\,{\text{normal}}\,{\text{subgroup}}\,{\text{of}}\,Z.

• A) {\text{True}}
• B) {\text{False}}
• C)
• D)

{\text{Let G be a group and H be a subgroup of G then H is normal iff }}\left( {{\text{aH}}} \right)\left( {{\text{bH}}} \right) \ne \left( {{\text{ab}}} \right){\text{H}},{\text{ for all a}},{\text{b}} \in {\text{G}}.

• A) True
• B) False
• C)
• D)

{\text{The}}\,{\text{factor}}\,{\text{group}}\,Z/nZ\,{\text{is}}\,{\text{isomorphic}}\,{\text{to}}\,{Z_n}

• A) {\text{True}}
• B) {\text{False}}
• C)
• D)

{\text{A homomorphism }}\varphi :G \to G'\,is{\text{ injective iff }}\ker {\text{(}}\varphi ) = \left\{ e \right\}{\text{.}}

• A) True
• B) False
• C)
• D)

{\text{If}}\,G\,{\text{is}}\,{\text{a}}\,{\text{finite}}\,{\text{group}}\,{\text{and}}\,N\,{\text{is}}\,{\text{a}}\,{\text{proper}}\,{\text{normal}}\,{\text{subgroup}}\,{\text{of}}\,G\,{\text{then}}\,G/N\,{\text{has}}\,{\text{the}}\,{\text{same}}\,{\text{structure}}\,{\text{as}}\,G.

• A) {\text{True}}
• B) {\text{false}}
• C)
• D)

\[{\text{Let G be a group and H be a subgroup of G then H is normal iff }}\left( {{\text{aH}}} \right)\left( {{\text{bH}}} \right) \ne \left( {{\text{ab}}} \right){\text{H}},{\text{ for all a}},{\text{b}} \in {\text{G}}.\]

• A) False
• B) True
• C)
• D)

{\text{The}}\,{\text{trivial}}\,{\text{subgroup}}\,N = \{ 0\} \,{\text{of}}\,Z\,{\text{is}}\,{\text{a}}\,{\text{normal}}\,{\text{subgroup}}\,{\text{of}}\,Z.

• A) {\text{True}}
• B) {\text{False}}
• C)
• D)