4. Find all of the composition series for each of the following groups. $$ \begin{array}{ll} ext { a. } \mathbb{Z}_{12} & ext { b. } \mathbb{Z}_{48} \ ext { c. The quaternions, } Q_{8} & ext { d. } D_{4} \ ext { e. } S_{3} imes \mathbb{Z}_{4} & ext { f. } S_{4} \ ext { g. } S_{n}, n \geq 5 & ext { h. } \mathbb{Q}\end{array} $$

Question

4. Find all of the composition series for each of the following groups. $$ \begin{array}{ll}	ext { a. } \mathbb{Z}_{12} & 	ext { b. } \mathbb{Z}_{48} \ 	ext { c. The quaternions, } Q_{8} & 	ext { d. } D_{4} \ 	ext { e. } S_{3} 	imes \mathbb{Z}_{4} & 	ext { f. } S_{4} \ 	ext { g. } S_{n}, n \geq 5 & 	ext { h. } \mathbb{Q}\end{array} $$

UpStudy AI · Accepted Answer

To find the composition series for each group, we need to analyze their structure and identify their normal subgroups. A composition series is a series of subgroups where each is normal in the next, and the factor groups are simple (having no nontrivial normal subgroups).
The group $$ \mathbb{Z}_{12} $$ has the following composition series:
1. $$ \{0\} 	riangleleft \mathbb{Z}_{4} 	riangleleft \mathbb{Z}_{12} $$
2. Factor groups: $$ \mathbb{Z}_{12}/\mathbb{Z}_{4} \cong \mathbb{Z}_{3} $$ and $$ \mathbb{Z}_{4}/\{0\} \cong \mathbb{Z}_{4} $$ (both simple).

Thus, the composition series is $$ \{0\} 	riangleleft \mathbb{Z}_{4} 	riangleleft \mathbb{Z}_{12} $$.
The group $$ \mathbb{Z}_{48} $$ has the following composition series:
1. $$ \{0\} 	riangleleft \mathbb{Z}_{16} 	riangleleft \mathbb{Z}_{48} $$
2. Factor groups: $$ \mathbb{Z}_{48}/\mathbb{Z}_{16} \cong \mathbb{Z}_{3} $$ and $$ \mathbb{Z}_{16}/\{0\} \cong \mathbb{Z}_{16} $$ (both simple).

Thus, the composition series is $$ \{0\} 	riangleleft \mathbb{Z}_{16} 	riangleleft \mathbb{Z}_{48} $$.
The group $$ Q_{8} $$ has the following composition series:
1. $$ \{1\} 	riangleleft \mathbb{Z}_{2} 	riangleleft Q_{8} $$
2. Factor groups: $$ Q_{8}/\mathbb{Z}_{2} \cong \mathbb{Z}_{4} $$ and $$ \mathbb{Z}_{2}/\{1\} \cong \mathbb{Z}_{2} $$ (both simple).

Thus, the composition series is $$ \{1\} 	riangleleft \mathbb{Z}_{2} 	riangleleft Q_{8} $$.
The group $$ D_{4} $$ has the following composition series:
1. $$ \{e\} 	riangleleft \mathbb{Z}_{2} 	riangleleft D_{4} $$
2. Factor groups: $$ D_{4}/\mathbb{Z}_{2} \cong \mathbb{Z}_{2} $$ and $$ \mathbb{Z}_{2}/\{e\} \cong \mathbb{Z}_{2} $$ (both simple).

Thus, the composition series is $$ \{e\} 	riangleleft \mathbb{Z}_{2} 	riangleleft D_{4} $$.
The group $$ S_{3} 	imes \mathbb{Z}_{4} $$ has the following composition series:
1. $$ \{e\} 	riangleleft \mathbb{Z}_{2} 	riangleleft S_{3} $$
2. Factor groups: $$ S_{3}/\mathbb{Z}_{2} \cong \mathbb{Z}_{3} $$ and $$ \mathbb{Z}_{2}/\{e\} \cong \mathbb{Z}_{2} $$ (both simple).

Thus, the composition series is $$ \{e\} 	riangleleft \mathbb{Z}_{2} 	riangleleft S_{3} 	imes \mathbb{Z}_{4} $$.
The group $$ S_{4} $$ has the following composition series:
1. $$ A_{4} 	riangleleft S_{4} $$
2. Factor groups: $$ S_{4}/A_{4} \cong \mathbb{Z}_{2} $$ and $$ A_{4} $$ has a composition series with $$ \{e\} 	riangleleft \mathbb{Z}_{3} $$.

Thus, the composition series is $$ \{e\} 	riangleleft A_{4} 	riangleleft S_{4} $$.
The group $$ S_{n} $$ for $$ n \geq 5 $$ has the following composition series:
1. $$ A_{n} 	riangleleft S_{n} $$
2. Factor groups: $$ S_{n}/A_{n} \cong \mathbb{Z}_{2} $$ and $$ A_{n} $$ has a composition series with $$ \{e\} $$.

Thus, the composition series is $$ \{e\} 	riangleleft A_{n} 	riangleleft S_{n} $$.
The group $$ \mathbb{Q} $$ is not a finite group and does not have a composition series in the traditional sense used for finite groups. However, it can be considered as a vector space over $$ \mathbb{Z} $$ with no nontrivial normal subgroups.

Thus, $$ \mathbb{Q} $$ does not have a composition series.
The composition series for the groups are as follows: - a. $$ \{0\} 	riangleleft \mathbb{Z}_{4} 	riangleleft \mathbb{Z}_{12} $$ - b. $$ \{0\} 	riangleleft \mathbb{Z}_{16} 	riangleleft \mathbb{Z}_{48} $$ - c. $$ \{1\} 	riangleleft \mathbb{Z}_{2} 	riangleleft Q_{8} $$ - d. $$ \{e\} 	riangleleft \mathbb{Z}_{2} 	riangleleft D_{4} $$ - e. $$ \{e\} 	riangleleft \mathbb{Z}_{2} 	riangleleft S_{3} 	imes \mathbb{Z}_{4} $$ - f. $$ \{e\} 	riangleleft A_{4} 	riangleleft S_{4} $$ - g. $$ \{e\} 	riangleleft A_{n} 	riangleleft S_{n} $$ for $$ n \geq 5 $$ - h. $$ \mathbb{Q} $$ does not have a composition series.

Find all of the composition series for each of the following groups.

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Other Questions