---- > [!definition] Definition. ([[subgroup]]) > If $G$ is a [[group]], a **subgroup** $H$ of $G$ is a subset of $G$ for which > 1. $e \in H$; > 2. $H$ is closed under $Gs [[group]] operation; > 3. $H$ is closed under taking inverses: $a \in H \implies a ^{-1} \in H$. > > [!example] Example. (Subgroups of $S_{3}$) > The subgroups of $S_{3}$ are exactly $\{ e \}$, $\{ e, \sigma \}, \{ e, \sigma \tau \}, \{ e, \sigma \tau^{2} \}, \{ e, \tau, \tau^{2} \}, S_{3}$. (The trick was to look at individual elements and their powers.) To see that these are the only subgroups, suppose that $H$ is a subgroup. Then $H$ must contain at least one of the listed [[group]]s. In every case, adding an element to one of these groups requires adding elements so that the group becomes $S_{3}$ (try it!). > ^d58546 ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```