MTH312 Tma Solutions
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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