8 Groups \(A , B , C\) and \(D\) are defined as follows:
\(A\) : the set of numbers \(\{ 2,4,6,8 \}\) under multiplication modulo 10 ,
\(B\) : the set of numbers \(\{ 1,5,7,11 \}\) under multiplication modulo 12 ,
\(C\) : the set of numbers \(\left\{ 2 ^ { 0 } , 2 ^ { 1 } , 2 ^ { 2 } , 2 ^ { 3 } \right\}\) under multiplication modulo 15,
\(D\) : the set of numbers \(\left\{ \frac { 1 + 2 m } { 1 + 2 n } \right.\), where \(m\) and \(n\) are integers \(\}\) under multiplication.
- Write down the identity element for each of groups \(A , B , C\) and \(D\).
- Determine in each case whether the groups
$$\begin{aligned}
& A \text { and } B ,
& B \text { and } C ,
& A \text { and } C
\end{aligned}$$
are isomorphic or non-isomorphic. Give sufficient reasons for your answers. - Prove the closure property for group \(D\).
- Elements of the set \(\left\{ \frac { 1 + 2 m } { 1 + 2 n } \right.\), where \(m\) and \(n\) are integers \(\}\) are combined under addition. State which of the four basic group properties are not satisfied. (Justification is not required.)
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