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

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

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

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

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

• A) False
• B) True
• C)
• D)

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)

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^{ - 1}}b \in H.

• A) True
• B) False
• C)
• D)

The\,quotient\,group\,R/Z\,is\,{\text{isomorphic}}\,{\text{to}}\,{\text{the circle}}\,{\text{group}}\,{\text{W = }}\left\{ {{{\text{e}}^\theta } \in C|\theta \in R} \right\}.

• A) False
• B) True
• C)
• D)

\[\begin{gathered} {\text{If }}\left( {{\text{G}}, \bullet } \right){\text{ and }}\left( {H,*} \right){\text{ are two groups}},{\text{ the function f:G}} \to {\text{H is called a group homomorphism if }} \hfill \\ {\text{f}}\left( {{\text{a}} \bullet {\text{b}}} \right){\text{ }} = {\text{ f}}\left( {\text{a}} \right) * {\text{f}}\left( {\text{b}} \right),{\text{ for all a}},{\text{b }} \in {\text{G}}. \hfill \\ \end{gathered} \]

• A) False
• B) True
• C)
• D)

\[\begin{gathered} {\text{If }}\left( {{\text{G}}, \bullet } \right){\text{ and }}\left( {H,*} \right){\text{ are two groups}},{\text{ the function f:G}} \to {\text{H is called a group homomorphism if }} \hfill \\ {\text{f}}\left( {{\text{a}} \bullet {\text{b}}} \right){\text{ }} \ne {\text{ f}}\left( {\text{a}} \right) * {\text{f}}\left( {\text{b}} \right),{\text{ for all a}},{\text{b }} \in {\text{G}}. \hfill \\ \end{gathered} \]

• A) True
• B) False
• C)
• D)

{\text{Let H be a subgroup of a group G}},{\text{ H is said to be a normal subgroup of G if}},{\text{ gH}} = {\text{Hg}},{\text{ }}\forall {\text{g }} \in {\text{G}}.

• A) False
• B) True
• C)
• D)

\begin{gathered} {\text{If }}\left( {{\text{G}}, \bullet } \right){\text{ and }}\left( {H,*} \right){\text{ are two groups}},{\text{ the function f:G}} \to {\text{H is called a group homomorphism if }} \hfill \\ {\text{f}}\left( {{\text{a}} \bullet {\text{b}}} \right){\text{ }} \ne {\text{ f}}\left( {\text{a}} \right) * {\text{f}}\left( {\text{b}} \right),{\text{ for all a}},{\text{b }} \in {\text{G}}. \hfill \\ \end{gathered}

• A) False
• B) True
• C)
• D)

\begin{gathered} {\text{Let K be a Kernal of the group morphism f:G}} \to {\text{H}}.{\text{ Then}}\,G/K\,{\text{is isomorphism to the image of}}\,f,{\text{ and the isomorphism}}{\text{is defined by}} \hfill \\ \psi {\text{: }}G/K \to \operatorname{Im} \,f\,is\,defined\,by\,\psi (Kg) = f(g). \hfill \\ \end{gathered}

• A) False
• B) True
• C)
• D)

The\,quotient\,group\,R/Z\,is\,{\text{isomorphic}}\,{\text{to}}\,{\text{the circle}}\,{\text{group}}\,{\text{W = }}\left\{ {{{\text{e}}^{i\theta }} \in C|\theta \in R} \right\}.

• A) False
• B) True
• C)
• D)

\[\begin{gathered} {\text{Let K be a Kernal of the group morphism f:G}} \to {\text{H}}.{\text{ Then}}\,G/K\,{\text{is isomorphism to the image of}}\,f,{\text{ and the isomorphism}}{\text{is defined by}} \hfill \\ \psi {\text{: }}G/K \to \operatorname{Im} \,f\,is\,defined\,by\,\psi (Kg) = f(g). \hfill \\ \end{gathered} \]

• A) False
• B) True
• C)
• D)

\begin{gathered} {\text{If }}\left( {{\text{G}}, \bullet } \right){\text{ and }}\left( {H,*} \right){\text{ are two groups}},{\text{ the function f:G}} \to {\text{H is called a group homomorphism if }} \hfill \\ {\text{f}}\left( {{\text{a}} \bullet {\text{b}}} \right){\text{ }} = {\text{ f}}\left( {\text{a}} \right) * {\text{f}}\left( {\text{b}} \right),{\text{ for all a}},{\text{b }} \in {\text{G}}. \hfill \\ \end{gathered}

• A) False
• B) True
• C)
• D)

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^{ - 1}}{b^{ - 1}} \in H.

• A) True
• B) False
• C)
• D)

\[The\,quotient\,group\,R/Z\,is\,{\text{isomorphic}}\,{\text{to}}\,{\text{the circle}}\,{\text{group}}\,{\text{W = }}\left\{ {{{\text{e}}^{i\theta }} \in C|\theta \in R} \right\}.\]

• A) False
• B) True
• C)
• D)

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

• A) True
• B) False
• C)
• D)

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

• A) False
• B) True
• C)
• D)

\[{\text{Let H be a subgroup of a group G}},{\text{ H is said to be a normal subgroup of G if}},{\text{ gH}} = {\text{Hg}},{\text{ }}\forall {\text{g }} \in {\text{G}}.\]

• A) True
• B) False
• C)
• D)

\[The\,quotient\,group\,R/Z\,is\,{\text{isomorphic}}\,{\text{to}}\,{\text{the circle}}\,{\text{group}}\,{\text{W = }}\left\{ {{{\text{e}}^\theta } \in C|\theta \in R} \right\}.\]

• A) False
• B) True
• C)
• D)