---- > [!theorem] Theorem. ([[classification of finite simple groups]]) > Every finite [[simple group]] is of one of four types: > 1. The [[cyclic group]] $C_{p}$ for $p$ a [[prime number|prime]]; > 2. The [[alternating group]] $A_{n}$ for $n \geq 6$; > 3. A finite [[simple group]] of [[lie group|lie]] type: these break into further subfamilies, e.g. the [[projective special linear group]] over a finite [[field]] (these are simple for all finite [[field]]s $\mathbb{F}$ when $n \geq 3$ and for all finite [[field]]s $\mathbb{F}$ with at least $4$ elements if $n=2$). > 4. One of $26$ *sporadic groups*. The largest of these is the monster $M$ and has order of about $8 \times 10^{53}.$ > [!proof]- Proof. (Classification of finite simple groups) > Left as an exercise. ---- #### ----- #### References > [!bac[](field.md)field.md)taview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as 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 > ```