3. Find all the subgroups of the quaternion group, $$ Q_{8} $$. Which subgroups are normal? What are all the factor groups of $$ Q_{8} $$ up to isomorphism?

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The quaternion group $$ Q_{8} $$ consists of the elements $$ \{ 1, -1, i, -i, j, -j, k, -k \} $$ with the following multiplication rules: 
- $$ i^2 = j^2 = k^2 = ijk = -1 $$.

To find the subgroups, we can consider the orders of the elements in $$ Q_{8} $$:
- The identity $$ \{ 1 \} $$ is a subgroup.
- The subgroup of order 2: $$ \{ 1, -1 \} $$.
- The subgroups of order 4: 
  - $$ \{ 1, -1, i, -i \} $$
  - $$ \{ 1, -1, j, -j \} $$
  - $$ \{ 1, -1, k, -k \} $$
- The whole group $$ Q_{8} $$ itself.

Thus, the subgroups of $$ Q_{8} $$ are:
1. $$ \{ 1 \} $$
2. $$ \{ 1, -1 \} $$
3. $$ \{ 1, -1, i, -i \} $$
4. $$ \{ 1, -1, j, -j \} $$
5. $$ \{ 1, -1, k, -k \} $$
6. $$ Q_{8} $$

In $$ Q_{8} $$, the normal subgroups are those that are invariant under conjugation. The normal subgroups are:
1. $$ \{ 1 \} $$
2. $$ \{ 1, -1 \} $$
3. $$ Q_{8} $$

The subgroups $$ \{ 1, -1, i, -i \} $$, $$ \{ 1, -1, j, -j \} $$, and $$ \{ 1, -1, k, -k \} $$ are not normal because conjugating elements like $$ i $$ by $$ j $$ or $$ k $$ does not yield elements within those subgroups.

The factor groups of $$ Q_{8} $$ can be determined by taking the quotient of $$ Q_{8} $$ by its normal subgroups:
1. $$ Q_{8}/\{ 1 \} \cong Q_{8} $$
2. $$ Q_{8}/\{ 1, -1 \} \cong \mathbb{Z}/2\mathbb{Z} $$ (the group with two elements)
3. $$ Q_{8}/Q_{8} \cong \{ 1 \} $$

Thus, the factor groups of $$ Q_{8} $$ up to isomorphism are:
1. $$ Q_{8} $$
2. $$ \mathbb{Z}/2\mathbb{Z} $$
3. $$ \{ 1 \} $$

The subgroups of the quaternion group $$ Q_{8} $$ are: 1. $$ \{ 1 \} $$ 2. $$ \{ 1, -1 \} $$ 3. $$ \{ 1, -1, i, -i \} $$ 4. $$ \{ 1, -1, j, -j \} $$ 5. $$ \{ 1, -1, k, -k \} $$ 6. $$ Q_{8} $$ The normal subgroups are: 1. $$ \{ 1 \} $$ 2. $$ \{ 1, -1 \} $$ 3. $$ Q_{8} $$ The factor groups of $$ Q_{8} $$ up to isomorphism are: 1. $$ Q_{8} $$ 2. $$ \mathbb{Z}/2\mathbb{Z} $$ 3. $$ \{ 1 \} $$

Find all the subgroups of the quaternion group, . Which subgroups are normal?
What are all the factor groups of up to isomorphism?

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Find all the subgroups of the quaternion group, . Which subgroups are normal? What are all the factor groups of up to isomorphism?