# Physics Notes by Jakob Schwichtenberg

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group_theory:notions:subgroups

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 group_theory:notions:subgroups [2017/01/07 13:12]jakobadmin [Subgroups] group_theory:notions:subgroups [2017/12/06 09:33] (current) Both sides previous revision Previous revision 2017/01/07 13:13 jakobadmin [Subgroups] 2017/01/07 13:12 jakobadmin [Subgroups] 2017/01/07 13:12 jakobadmin [Subgroups] 2017/01/07 13:12 jakobadmin [Subgroups] 2017/01/07 13:08 jakobadmin [Normal Subgroups] 2017/01/07 13:07 jakobadmin [Normal Subgroups] 2017/01/07 13:07 jakobadmin created Next revision Previous revision 2017/01/07 13:13 jakobadmin [Subgroups] 2017/01/07 13:12 jakobadmin [Subgroups] 2017/01/07 13:12 jakobadmin [Subgroups] 2017/01/07 13:12 jakobadmin [Subgroups] 2017/01/07 13:08 jakobadmin [Normal Subgroups] 2017/01/07 13:07 jakobadmin [Normal Subgroups] 2017/01/07 13:07 jakobadmin created Line 3: Line 3: A subgroup $H$ of a given group $G$ consists of elements of $G$ that have some additional property. ​ 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 condition ​$M^T M = 1$ **and** $\det(M) =1$.) + 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 The mathematical notation to indicate that some group $H$ is a subgroup of another group $G$ is