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

Normal subgroups are those that are invariant under conjugation. The normal subgroups of $$ Q_{8} $$ are:
1. $$ \{ 1 \} $$  
2. $$ \{ 1, -1 \} $$  
3. $$ Q_{8} $$ itself  
4. $$ \{ 1, -1, i, -i \} $$  
5. $$ \{ 1, -1, j, -j \} $$  
6. $$ \{ 1, -1, k, -k \} $$

The factor groups of $$ Q_{8} $$ up to isomorphism are obtained by taking the quotient of $$ Q_{8} $$ by its normal subgroups:
1. $$ Q_{8} / \{ 1 \} \cong Q_{8} $$  
2. $$ Q_{8} / \{ 1, -1 \} \cong \mathbb{Z}_{2} 	imes \mathbb{Z}_{2} $$  
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. $$ Q_{8} / Q_{8} \cong \{ 1 \} $$

The subgroups of $$ Q_{8} $$ are $$ \{ 1 \}, \{ 1, -1 \}, \{ 1, i \}, \{ 1, -i \}, \{ 1, j \}, \{ 1, -j \}, \{ 1, k \}, \{ 1, -k \}, \{ 1, -1, i, -i \}, \{ 1, -1, j, -j \}, \{ 1, -1, k, -k \}, \{ 1, i, j, k \}, Q_{8} $$. The normal subgroups are $$ \{ 1 \}, \{ 1, -1 \}, Q_{8}, \{ 1, -1, i, -i \}, \{ 1, -1, j, -j \}, \{ 1, -1, k, -k \} $$. 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?