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advanced_tools:group_theory:subgroup [2017/12/17 10:48] (current)
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 +====== 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>​
 +
  
advanced_tools/group_theory/subgroup.txt ยท Last modified: 2017/12/17 10:48 (external edit)