----- > [!proposition] Proposition. ([[the Sylow Test for Nonsimplicity]]) > Let $n \in \mathbb{N}$ be [[prime number|nonprime]], and let $p$ be a [[prime number|prime]] [[divides|divisor]] of $n$. If $1$ is the only divisor of $n$ [[congruent]] to $1$ modulo $p$, there there does not exist a [[simple group]] of order $n$. ^0c4966 > [!proof]- Proof. ([[the Sylow Test for Nonsimplicity]]) > If $|G|=p^{n}$ for some [[prime number|prime]] $p$, then by [[p-groups have nontrivial centers]] and that [[center of a group|centers]] are [[normal subgroup]]s $G$ is not simple. Else $|G|=p^{n}m$ for some $m>1$, in which case $G$ has a proper [[p-Sylow subgroup]] of [[order of an element in a group|order]] $p^{n}$. This [[p-Sylow subgroup]] is [[normal subgroup|normal]] if it is the only one; by the [[the Sylow theorems|third Sylow theorem]] this happens when $1$ is the only divisor of $n$ [[congruent]] to $1$ modulo $p$. ^a79def ----- #### ---- #### 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 > ```