----- > [!proposition] Proposition. ([[p-group has nontrivial center]]) > Let $G$ be a [[group]] and $p$ a [[prime number]], with $|G|=p^{n}$. The the [[center of a group|center]] $Z(G)$ of $G$ is nontrivial: $Z(G)>e$. > [!proof]- Proof. ([[p-group has nontrivial center]]) > Suppose $x \notin Z(G)$. $Z_{G}(x)$ is a [[subgroup]] of $G$ (with order at least 2) and thus by [[Lagrange's Theorem]] $|Z_{G}(x)| \ \b | \ |G|$. Therefore, because all [[divides|divisors]] of $p^{n}$ are powers of $p$, we have $|Z_{G}(x)|=p^{i}$ for some $i < |G|$. This implies $\frac{|G|}{|Z_{G}(x)|}=\frac{p^{n}}{p^{i}}$ is [[divides|divisible]] by $p$. \ Now, if $x \notin Z(G)$, then $1<|Z_{G}(x)| < |Z(G)|$. Since both are [[subgroup|subgroups]] this enforces that $|Z(G)|$ is a multiple of $p$. But this means $p | |Z(G)|$, and so by [[Cauchy's Theorem]] we discover that $Z(G)$ contains an element of order $p$ and thus $Z(G)>e$. ----- #### ---- #### 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 > ```