====== Subgroups ======

<tabbox Why is it interesting?> 

<tabbox Layman> 

<note tip>
Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
</note>
  
<tabbox Student> 

A subgroup $H$ of a given group $G$ consists of elements of $G$ that have some additional property. 

For example, the subgroup $SO(N)$ of $O(N)$ consists of all $N \times N$ matrices with determinant equal to $1$. ($O(N)$ consists of all $N \times N$ matrices $M$ that fulfil the condition $M^T M = 1$. $SO(N)$ consists of all $N \times N$ matrices $M$ that fulfil the conditions $M^T M = 1$ **and** $\det(M) =1$.)

The mathematical notation to indicate that some group $H$ is a subgroup of another group $G$ is 

$$ H \subset G .$$

** Normal Subgroups:**

<blockquote>
[A] normal subgroup [is] a subgroup that "looks the same from every perspective." For example, the subgroup of translations in the Euclidean group is always normal because the description "$g$ is a translation" is the same from every perspective (that is, it's invariant under conjugation).

<cite>http://math.stackexchange.com/a/11976/120960</cite>
</blockquote>
 
<tabbox Researcher> 

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

  
<tabbox Examples> 

--> Example1#

 
<--

--> Example2:#

 
<--

<tabbox FAQ> 
  
<tabbox History> 

</tabbox>