advanced_tools:group_theory:subgroup

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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 | ||

+ | |||

+ | $$ H \subset G .$$ | ||

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+ | ** 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). | ||

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+ | <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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