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 possible orders of subgroups are 1, 2, 4, and 8. The subgroups are as follows:

1. The trivial subgroup: $$ \{ 1 \} $$
2. The subgroup of order 2: $$ \{ 1, -1 \} $$
3. The subgroups of order 4:
   - $$ \{ 1, -1, i, -i \} $$
   - $$ \{ 1, -1, j, -j \} $$
   - $$ \{ 1, -1, k, -k \} $$
4. The whole group: $$ Q_{8} = \{ 1, -1, i, -i, j, -j, k, -k \} $$

Thus, the subgroups of $$ Q_{8} $$ are:
- $$ \{ 1 \} $$
- $$ \{ 1, -1 \} $$
- $$ \{ 1, -1, i, -i \} $$
- $$ \{ 1, -1, j, -j \} $$
- $$ \{ 1, -1, k, -k \} $$
- $$ Q_{8} $$
Normal subgroups are those that are invariant under conjugation by any element of the group. In $$ Q_{8} $$, the normal subgroups are:
- $$ \{ 1 \} $$
- $$ \{ 1, -1 \} $$
- $$ \{ 1, -1, i, -i \} $$
- $$ \{ 1, -1, j, -j \} $$
- $$ \{ 1, -1, k, -k \} $$
- $$ Q_{8} $$

All subgroups of $$ Q_{8} $$ are normal because $$ Q_{8} $$ is a group of order 8, and all subgroups of index 2 are normal.
The factor groups of $$ Q_{8} $$ can be determined by taking the quotient of $$ Q_{8} $$ by its normal subgroups. The factor groups are:
1. $$ Q_{8} / \{ 1 \} \cong Q_{8} $$
2. $$ Q_{8} / \{ 1, -1 \} \cong \mathbb{Z}_{2} 	imes \mathbb{Z}_{2} $$
3. $$ Q_{8} / \{ 1, -1, i, -i \} \cong \mathbb{Z}_{2} 	imes \mathbb{Z}_{2} $$
4. $$ Q_{8} / \{ 1, -1, j, -j \} \cong \mathbb{Z}_{2} 	imes \mathbb{Z}_{2} $$
5. $$ Q_{8} / \{ 1, -1, k, -k \} \cong \mathbb{Z}_{2} 	imes \mathbb{Z}_{2} $$
6. $$ Q_{8} / Q_{8} \cong \{ 1 \} $$

Thus, the factor groups of $$ Q_{8} $$ up to isomorphism are:
- $$ Q_{8} $$
- $$ \mathbb{Z}_{2} 	imes \mathbb{Z}_{2} $$
- $$ \{ 1 \} $$
The subgroups of the quaternion group $$ Q_{8} $$ are: - $$ \{ 1 \} $$ - $$ \{ 1, -1 \} $$ - $$ \{ 1, -1, i, -i \} $$ - $$ \{ 1, -1, j, -j \} $$ - $$ \{ 1, -1, k, -k \} $$ - $$ Q_{8} $$ All subgroups are normal. The factor groups of $$ Q_{8} $$ up to isomorphism are: - $$ Q_{8} $$ - $$ \mathbb{Z}_{2} 	imes \mathbb{Z}_{2} $$ - $$ \{ 1 \} $$

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?