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

• A) False
• B) True
• C)
• D)

A bijective group homomorphism is called a group isomorphism.

• A) True
• B) False
• C)
• D)

Every subgroup H of an nonabelian group G is normal.

• A) True
• B) False
• C)
• D)

The following are three equivalent conditions for a subgroup H of a group G to be a normal subgroup of G: 1. ghg-1∊H for all g∊G and h∊H. 2. gHg-1=H for all g∊G. 3. gH=Hg for all g∊G.

• A) True
• B) False
• C)
• D)

A factor group of a cyclic group is cyclic.

• 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) True
• B) False
• 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)

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

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

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

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

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

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

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

• A) $${\text{False}}$$
• B) $${\text{True}}$$
• 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) $${Z_2}$$
• 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)