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 — advanced_tools:group_theory:subgroup [2017/12/17 10:48] (current) 2017/12/17 11:48 jakobadmin created 2017/12/17 11:48 jakobadmin created Line 1: Line 1: + ====== Subgroups ====== + +  ​ + +  ​ + + + 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. + ​ + ​ +  ​ + + 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​ + ​ + +  ​ + + + The motto in this section is: //the higher the level of abstraction,​ the better//. + ​ + + ​ +  ​ + + --> Example1# + + + <-- + + --> Example2:# + + + <-- + +  ​ + ​ +  ​ + + ​ +