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.

1. Write down the identity element for each of groups \(A , B , C\) and \(D\).
2. 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.
3. Prove the closure property for group \(D\).
4. 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.) \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }