1. A ring that is commutative ring and has a multiplicative ring with unity is called

a kernel commutative ring

an identity commutative ring with euclidean property

a unique commutative ring with one element

—>> a commutative ring with unity

2. A subgroup \(H\) of a group \(G\) is normal if

\(Hx=xH\) for some \(x\in G\)

\(Hx\neq xH\) for all \(x\in G\)

—>> \(H/xH\) for some \(x\in G\)

\(Hx=G\) for all \(x\in G\)

3. Let \(R\) be a ring and \(I_R\) be the identity map then \(ker I_R\) is

\(R\)

—>> \({0}\)

\frac{R}{I_R}

empty

4. Given groups \(G\) and \(G^{\ast}\), a homomorphism \(\theta : G \rightarrow G^{\ast}\) is called an isomorphism if

\(\theta\) is surjective and inclusive

—>> \(\theta\) is injective and surjective

\(\theta\) is normal in \(G\)

\(\theta\) is inclusive and injective

5. The following product of cycles are the same except

\((1 3 2 4)\)

\((4 1 3 2)\)

\((2 4 1 3)\)

—>> \((1 2 3 4)\)

6. Given that \(A\) and \(B\) are cyclic groups of order \(m\) and \(n\) respectively, then \(A\times B\) is a

a normal subgroup of oder \(\frac{m}{n}\)

a permutation group of oder \(m+n\)

—>> a cyclic group of oder \(mn\)

a Cayley’s subgroup of oder \(m-n\)

7. If \(H\) and \(K\) are subgroups of a group \(G\) with \(K\) normal in \(G\), then

—>> \(H/(H\cap K)\)\cong (HK)/K\)

\(H/(H\cap K)\)\cong K/(HK)\)

\(H/(H\cap K)\)\cong H\cap K)/K\)

\(H/(H\cap K)\)\cong K/H\cap K\)

8. The index of a normal subgroup of a group \(G\) is

1

—>> 2

3

\(0\)

9. Given a permutation \(g=\left(\begin{array}{ccccc}1&2&3&4&5\\3&5&4&1&2\end{array}\right)\), then

—>> \(g=(1 3 4)(2 5)\)

\(g=(1 4 2)(3 5)\)

\(g=(1 2 3 4 5)\)

\(g=(1 3 4 2)\)

10. \(Im(sign)=\)

—>> \({1, -1}\)

\({1}\)

\({-1}\)

\({0,1}\)

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—>> \({1, -1}\)

\({1}\)

\({-1}\)

\({0,1}\)

—>> \(g=(1 3 4)(2 5)\)

\(g=(1 4 2)(3 5)\)

\(g=(1 2 3 4 5)\)

\(g=(1 3 4 2)\)

1

—>> 2

3

\(0\)

—>> \(H/(H\cap K)\)\cong (HK)/K\)

\(H/(H\cap K)\)\cong K/(HK)\)

\(H/(H\cap K)\)\cong H\cap K)/K\)

\(H/(H\cap K)\)\cong K/H\cap K\)

a normal subgroup of oder \(\frac{m}{n}\)

a permutation group of oder \(m+n\)

—>> a cyclic group of oder \(mn\)

a Cayley’s subgroup of oder \(m-n\)

\((1 3 2 4)\)

\((4 1 3 2)\)

\((2 4 1 3)\)

—>> \((1 2 3 4)\)

\(\theta\) is surjective and inclusive

—>> \(\theta\) is injective and surjective

\(\theta\) is normal in \(G\)

\(\theta\) is inclusive and injective

a kernel commutative ring

an identity commutative ring with euclidean property

a unique commutative ring with one element

—>> a commutative ring with unity

\(Hx=xH\) for some \(x\in G\)

\(Hx\neq xH\) for all \(x\in G\)

—>> \(H/xH\) for some \(x\in G\)

\(Hx=G\) for all \(x\in G\)