Editing Simple group (section)

== Classification ==
There is as yet no known classification for general (infinite) simple groups, and no such classification is expected. One reason for this is the existence of continuum-many [[Tarski monster group]]s for every sufficiently-large prime characteristic, each simple and having only the cyclic group of that characteristic as its subgroups.<ref>{{citation|contribution-url=https://www.raczar.es/webracz/ImageServlet?mod=publicaciones&subMod=monografias&car=monografia26&archivo=089Otal.pdf|contribution=The Classification of the Finite Simple Groups: An Overview|first=Javier|last=Otal|title=Problemas del Milenio |editor-first=L. J.|editor-last=Boya|series=Monografías de la Real Academia de Ciencias Exactas, Físicas, Químicas y Naturales de Zaragoza|volume=26|publisher=Real Academia de Ciencias Exactas, Físicas, Químicas y Naturales de Zaragoza|year=2004}}</ref>

=== Finite simple groups ===
{{main|list of finite simple groups}}
{{further|Classification of finite simple groups}}
The [[List of finite simple groups|finite simple groups]] are important because in a certain sense they are the "basic building blocks" of all finite groups, somewhat similar to the way [[prime number]]s are the basic building blocks of the [[integer]]s. This is expressed by the [[Jordan–Hölder theorem]] which states that any two [[composition series]] of a given group have the same length and the same factors, [[up to]] [[permutation]] and [[isomorphism]]. In a huge collaborative effort, the [[classification of finite simple groups]] was declared accomplished in 1983 by [[Daniel Gorenstein]], though some problems surfaced (specifically in the classification of [[quasithin group]]s, which were plugged in 2004).

Briefly, finite simple groups are classified as lying in one of 18 families, or being one of 26 exceptions:
* <math>\mathbb{Z}_p</math> – [[cyclic group]] of prime order
* <math>A_n</math> – [[alternating group]] for <math>n\geq5</math>
*:The alternating groups may be considered as groups of Lie type over the [[field with one element]], which unites this family with the next, and thus all families of non-abelian finite simple groups may be considered to be of Lie type.
* One of 16 families of [[groups of Lie type]] or their derivatives
*:The [[Tits group]] is generally considered of this form, though strictly speaking it is not of Lie type, but rather index 2 in a group of Lie type.
* One of 26 exceptions, the [[sporadic group]]s, of which 20 are subgroups or [[subquotient]]s of the [[monster group]] and are referred to as the "Happy Family", while the remaining 6 are referred to as [[pariah group|pariahs]].