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