---- > [!definition] Definition. ([[simple group]]) > A [[group]] $G$ is said to be **simple** if it contains no [[normal subgroup|normal subgroups]] other than $G$ and $(e)$. > [!basicexample] > - Any [[group]] of [[prime number|prime order]] ($C_{p}$) > - Finite [[abelian group]] if and only if it has [[prime number|prime order]] (iff it is $C_{p}$). This is because if not [[prime number|prime order]], then [[Cauchy's Theorem]] guarantees a [[subgroup]] of [[order of an element in a group|order]] $p$ for some $p$ [[divides|dividing]] the order, and all [[subgroup]]s of [[abelian group]]s are [[normal subgroup]]s. > - [[There are no simple nonabelian groups of order less than 60]] ---- #### ---- #### 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 > ```