Problem 1 (5 points) Viewing $$ \langle 2\rangle $$ and $$ \langle 8\rangle $$ as subgroups of the additive group $$ \mathbb{Z} $$ of the integers, $$ \langle 8\rangle $$ is a subgroup $$ \langle 2\rangle $$. Compute the addition table of the factor group $$ \langle 2\rangle /\langle 8\rangle $$. Hint. $$ \langle 2\rangle=\{2 t \mid t \in \mathbb{Z}\} $$ and $$ \langle 8\rangle=\{8 t \mid t \in \mathbb{Z}\} $$ are the cyclic subgroups of $$ \mathbb{Z} $$ generated by 2 and 8 respectively. First find which are the different cosets of $$ \langle 2\rangle /\langle 8\rangle=\{2 t+\langle 8\rangle \mid t \in \mathbb{Z}\} $$.

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To compute the addition table of the factor group $$ \langle 2\rangle /\langle 8\rangle $$, we first need to identify the cosets of $$ \langle 8\rangle $$ in $$ \langle 2\rangle $$.

1. **Identify the subgroups**:
   - The subgroup $$ \langle 2\rangle $$ consists of all integer multiples of 2:
     $$
     \langle 2\rangle = \{ 2t \mid t \in \mathbb{Z} \} = \{ \ldots, -4, -2, 0, 2, 4, 6, \ldots \}
     $$
   - The subgroup $$ \langle 8\rangle $$ consists of all integer multiples of 8:
     $$
     \langle 8\rangle = \{ 8t \mid t \in \mathbb{Z} \} = \{ \ldots, -16, -8, 0, 8, 16, \ldots \}
     $$

2. **Find the cosets of $$ \langle 8\rangle $$ in $$ \langle 2\rangle $$**:
   - The cosets of $$ \langle 8\rangle $$ in $$ \langle 2\rangle $$ can be represented as $$ 2t + \langle 8\rangle $$ for $$ t \in \mathbb{Z} $$.
   - We can compute a few cosets:
     - For $$ t = 0 $$:
       $$
       2 \cdot 0 + \langle 8\rangle = \langle 8\rangle = \{ 0, 8, -8, 16, -16, \ldots \}
       $$
     - For $$ t = 1 $$:
       $$
       2 \cdot 1 + \langle 8\rangle = 2 + \langle 8\rangle = \{ 2, 10, -6, 18, -14, \ldots \}
       $$
     - For $$ t = 2 $$:
       $$
       2 \cdot 2 + \langle 8\rangle = 4 + \langle 8\rangle = \{ 4, 12, -4, 20, -12, \ldots \}
       $$
     - For $$ t = 3 $$:
       $$
       2 \cdot 3 + \langle 8\rangle = 6 + \langle 8\rangle = \{ 6, 14, -2, 22, -10, \ldots \}
       $$

3. **Identify distinct cosets**:
   - Notice that the cosets repeat every 8 units because $$ \langle 8\rangle $$ has a period of 8. Thus, we can summarize the distinct cosets:
     - $$ \langle 8\rangle $$ (which corresponds to $$ 0 + \langle 8\rangle $$)
     - $$ 2 + \langle 8\rangle $$
     - $$ 4 + \langle 8\rangle $$
     - $$ 6 + \langle 8\rangle $$

Therefore, the distinct cosets of $$ \langle 2\rangle / \langle 8\rangle $$ are:
   $$
   \{ \langle 8\rangle, 2 + \langle 8\rangle, 4 + \langle 8\rangle, 6 + \langle 8\rangle \}
   $$

4. **Addition in the factor group**:
   - We can denote the cosets as:
     - $$ 0 $$ for $$ \langle 8\rangle $$
     - $$ 1 $$ for $$ 2 + \langle 8\rangle $$
     - $$ 2 $$ for $$ 4 + \langle 8\rangle $$
     - $$ 3 $$ for $$ 6 + \langle 8\rangle $$

Now we can compute the addition table for these cosets:
   $$
   \begin{array}{c|cccc}
     + & 0 & 1 & 2 & 3 \
     \hline
     0 & 0 & 1 & 2 & 3 \
     1 & 1 & 2 & 3 & 0 \
     2 & 2 & 3 & 0 & 1 \
     3 & 3 & 0 & 1 & 2 \
   \end{array}
   $$

5. **Conclusion**:
   The addition table for the factor group $$ \langle 2\rangle / \langle 8\rangle $$ is as follows:
   $$
   \begin{array}{c|cccc}
     + & 0 & 1 & 2 & 3 \
     \hline
     0 & 0 & 1 & 2 & 3 \
     1 & 1 & 2 & 3 & 0 \
     2 & 2 & 3 & 0 & 1 \
     3 & 3 & 0 & 1 & 2 \
   \end{array}
   $$
The distinct cosets of $$ \langle 2\rangle / \langle 8\rangle $$ are $$ \langle 8\rangle, 2 + \langle 8\rangle, 4 + \langle 8\rangle, 6 + \langle 8\rangle $$. The addition table for these cosets is: $$ \begin{array}{c|cccc} + & 0 & 1 & 2 & 3 \ \hline 0 & 0 & 1 & 2 & 3 \ 1 & 1 & 2 & 3 & 0 \ 2 & 2 & 3 & 0 & 1 \ 3 & 3 & 0 & 1 & 2 \ \end{array} $$

Problem 1 (5 points) Viewing and as subgroups of the additive group of the integers,
is a subgroup . Compute the addition table of the factor group .
Hint. and are the cyclic subgroups of generated by 2 and
8 respectively. First find which are the different cosets of .

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Problem 1 (5 points) Viewing and as subgroups of the additive group of the integers, is a subgroup . Compute the addition table of the factor group . Hint. and are the cyclic subgroups of generated by 2 and 8 respectively. First find which are the different cosets of .