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--- advanced_tools:group_theory:subgroup [2017/12/17 11:48]
+++ advanced_tools:group_theory:subgroup [2017/12/17 12:48]
jakobadmin created
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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>

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