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''finite simple group''

A [[Finite group|finite group]] without normal subgroups (cf. [[Normal subgroup|Normal subgroup]]) different from the trivial subgroup and the whole group. The finite simple groups are the smallest  "building blocks"  from which one can  "construct"  any finite group by means of extensions. Every factor of a [[Composition sequence|composition sequence]] of a finite group is a finite simple group, while a minimal normal subgroup is a direct product of finite simple groups. The cyclic groups of prime order are the easiest examples of finite simple groups. Only these finite simple groups occur as factors of composition sequences of solvable groups (cf. [[Solvable group|Solvable group]]). All other finite simple groups are non-solvable, and their orders are even (cf. [[Burnside problem|Burnside problem]] 1)). The alternating groups  $  \mathfrak A _ {n} $, 
the projective special linear groups  $   \mathop{\rm PSL} ( n , q ) $
over a finite field of order  $  q $, 
the projective symplectic groups  $   \mathop{\rm PSP} ( 2 n , q ) $, 
the projective orthogonal groups  $  \textrm{ P } \Omega ( n , q ) $, 
and the projective unitary groups  $   \mathop{\rm PSU} ( n , q  ^ {2} ) $
give an infinite number of examples of non-cyclic finite simple groups. All finite simple groups listed were already known in the 19th century. Besides these, at the end of that century  $  5 $
more groups were discovered (cf. [[Mathieu group|Mathieu group]]). At the beginning of the 20th century finite analogues of the simple Lie groups of type  $  G _ {2} $(
cf. [[Dickson group|Dickson group]]) were constructed. The discovery of new infinite series of finite simple groups, in the 1950s, made it possible to obtain the majority of types of known simple groups from automorphism groups of simple Lie algebras (cf. [[Chevalley group|Chevalley group]]). The known infinite series of finite simple groups are listed in the table below.<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1">Denotations, related to the type of the corresponding Lie algebras</td> <td colname="2" style="background-color:white;" colspan="1">Alternative denotations</td> <td colname="3" style="background-color:white;" colspan="1">Conditions of existence of a simple finite group</td> <td colname="4" style="background-color:white;" colspan="1">Order of the group</td> <td colname="5" style="background-color:white;" colspan="1"> $  d $
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"> $  \mathbf Z _ {p} $
</td> <td colname="3" style="background-color:white;" colspan="1"> $  p $
a prime number</td> <td colname="4" style="background-color:white;" colspan="1"> $  p $
</td> <td colname="5" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"> $  \mathfrak A _ {l} $
</td> <td colname="3" style="background-color:white;" colspan="1"> $  l \geq  5 $
</td> <td colname="4" style="background-color:white;" colspan="1"> $  l! / 2 $
</td> <td colname="5" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  A _ {l} ( q) $
</td> <td colname="2" style="background-color:white;" colspan="1">PSL $  ( l+ 1 , q ) $
</td> <td colname="3" style="background-color:white;" colspan="1"> $  l \geq  2 ;  l = 1 , q \geq  4 $
</td> <td colname="4" style="background-color:white;" colspan="1"> $  q ^ {l( l+ 1)/2 } ( q  ^ {2} - 1) \dots ( q  ^ {l+} 1 - 1) /d $
</td> <td colname="5" style="background-color:white;" colspan="1"> $  ( l+ 1 , q- 1 ) $
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  B _ {l} ( q) $
</td> <td colname="2" style="background-color:white;" colspan="1"> $  \textrm{ P } \Omega ( 2l+ 1, q) $
</td> <td colname="3" style="background-color:white;" colspan="1"> $  l \geq  3;  l= 2, q\geq  3; l= 1 , q \geq  4 $
</td> <td colname="4" style="background-color:white;" colspan="1"> $  q ^ {l  ^ {2} } ( q  ^ {2} - 1) \dots ( q  ^ {2l} - 1)/d $
</td> <td colname="5" style="background-color:white;" colspan="1"> $  ( 2 , q- 1) $
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  C _ {l} ( q) $
</td> <td colname="2" style="background-color:white;" colspan="1">PSP  $  ( 2l, q) $
</td> <td colname="3" style="background-color:white;" colspan="1"> $  l \geq  3;  l= 2, q \geq  3; l= 1 , q \geq  4 $
</td> <td colname="4" style="background-color:white;" colspan="1"> $  q ^ {l  ^ {2} } ( q  ^ {2} - 1) \dots ( q  ^ {2l} - 1)/d $
</td> <td colname="5" style="background-color:white;" colspan="1"> $  ( 2 , q- 1) $
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  D _ {l} ( q) $
</td> <td colname="2" style="background-color:white;" colspan="1">P  $  \Omega  ^ {+} ( 2l , q) $
</td> <td colname="3" style="background-color:white;" colspan="1"> $  l \geq  3 $
</td> <td colname="4" style="background-color:white;" colspan="1"> $  q ^ {l ( l- 1) } ( q  ^ {2} - 1) \dots ( q  ^ {2l-} 2 - 1)( q  ^ {l} - 1)/d $
</td> <td colname="5" style="background-color:white;" colspan="1"> $  ( 4 , q  ^ {l} - 1) $
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  E _ {6} ( q) $
</td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"> $  q  ^ {36} ( q  ^ {2} - 1 )( q  ^ {5} - 1) ( q  ^ {6} - 1) ( q  ^ {8} - 1 )( q  ^ {9} - 1 )( q  ^ {12} - 1) /d $
</td> <td colname="5" style="background-color:white;" colspan="1"> $  ( 3 , q- 1 ) $
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  E _ {7} ( q) $
</td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"> $  q  ^ {63} ( q  ^ {2} - 1 )( q  ^ {6} - 1) ( q  ^ {8} - 1) ( q  ^ {10} - 1 )( q  ^ {12} - 1 )( q  ^ {14} - 1) ( q  ^ {18} - 1 )/d $
</td> <td colname="5" style="background-color:white;" colspan="1"> $  ( 2 , q- 1) $
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  E _ {8} ( q) $
</td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"> $  q  ^ {120} ( q  ^ {2} - 1 )( q  ^ {8} - 1) ( q  ^ {12} - 1)( q  ^ {14} - 1 ) ( q  ^ {18} - 1) ( q  ^ {20} - 1 ) ( q  ^ {24} - 1 ) ( q  ^ {30} - 1) $
</td> <td colname="5" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  F _ {4} ( q) $
</td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"> $  q  ^ {24} ( q  ^ {2} - 1 )( q  ^ {6} - 1)( q  ^ {8} - 1)( q  ^ {12} - 1) $
</td> <td colname="5" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  G _ {2} ( q) $
</td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"> $  q  ^ {6} ( q  ^ {2} - 1 )( q  ^ {6} - 1 ) $
</td> <td colname="5" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  {}  ^ {2} A _ {l} ( q  ^ {2} ) $
</td> <td colname="2" style="background-color:white;" colspan="1">PSU  $  ( l+ 1, q  ^ {2} ) $
</td> <td colname="3" style="background-color:white;" colspan="1"> $  l\geq  3;  l= 2, q\geq  3; l = 1 , q \geq  4 $
</td> <td colname="4" style="background-color:white;" colspan="1"> $  q ^ {l( l+ 1)/2 } ( q  ^ {2} - 1 )( q  ^ {3} + 1 ) \dots ( q  ^ {l+} 1 + ( - 1 )  ^ {l} ) / d $
</td> <td colname="5" style="background-color:white;" colspan="1"> $  ( l+ 1 , q+ 1 ) $
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  {}  ^ {2} D _ {l} ( q  ^ {2} ) $
</td> <td colname="2" style="background-color:white;" colspan="1">P  $  \Omega  ^ {-} ( 2l , q ) $
</td> <td colname="3" style="background-color:white;" colspan="1"> $  l \geq  2 $
</td> <td colname="4" style="background-color:white;" colspan="1"> $  q ^ {l( l- 1 ) /2 } ( q  ^ {2} - 1 )( q  ^ {4} - 1) \dots ( q  ^ {2l-} 2 - 1)( q  ^ {l} + 1 ) /d $
</td> <td colname="5" style="background-color:white;" colspan="1"> $  ( 4 , q  ^ {l} + 1 ) $
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  {}  ^ {2} E _ {6} ( q  ^ {2} ) $
</td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"> $  q  ^ {36} ( q  ^ {2} - 1 ) ( q  ^ {5} + 1 )( q  ^ {6} - 1)( q  ^ {8} - 1)( q  ^ {9} + 1)( q  ^ {12} - 1) / d $
</td> <td colname="5" style="background-color:white;" colspan="1"> $  ( 3 , q+ 1) $
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  {}  ^ {3} D _ {4} ( q  ^ {3} ) $
</td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"></td> <td colname="4" style="background-color:white;" colspan="1"> $  q  ^ {12} ( q  ^ {2} - 1)( q  ^ {6} - 1)( q  ^ {8} + q  ^ {4} + 1 ) $
</td> <td colname="5" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  {}  ^ {2} B _ {2} ( q) $
</td> <td colname="2" style="background-color:white;" colspan="1"> $  Sz( q) $
</td> <td colname="3" style="background-color:white;" colspan="1"> $  q= 2  ^ {2l+} 1 $
</td> <td colname="4" style="background-color:white;" colspan="1"> $  q  ^ {2} ( q- 1)( q  ^ {2} + 1 ) $
</td> <td colname="5" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  {}  ^ {2} G _ {2} ( q) $
</td> <td colname="2" style="background-color:white;" colspan="1"> $  R( q) $
</td> <td colname="3" style="background-color:white;" colspan="1"> $  q= 3  ^ {3l+} 1 $
</td> <td colname="4" style="background-color:white;" colspan="1"> $  q  ^ {3} ( q- 1)( q  ^ {3} + 1) $
</td> <td colname="5" style="background-color:white;" colspan="1"></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  {}  ^ {2} F _ {4} ( q)  ^  \prime  $
</td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"> $  q = 2  ^ {2l-} 1 $
</td> <td colname="4" style="background-color:white;" colspan="1"> $  q  ^ {12} ( q- 1)( q  ^ {3} + 1)( q 4 - 1)( q  ^ {6} + 1)/d $
</td> <td colname="5" style="background-color:white;" colspan="1"> $  \begin{array}{c}
{2 \textrm{ for } q = 2 } \\
 {1 for q > 2 } 
\end{array}
$
</td> </tr> </tbody> </table>

</td></tr> </table>

Here,  $  q $
is a non-zero power of a prime number,  $  l $
is a natural number and  $  ( s , t ) $
is the greatest common divisor of two numbers  $  s $
and  $  t $. 
Apart from those in the table, 26 other finite simple groups are known; they do not fit in any infinite series of finite simple groups (the so-called sporadic simple groups, cf. [[Sporadic simple group|Sporadic simple group]]).

A basic problem in the theory of finite simple groups is the problem of classifying all of them. It consists of the proof that every finite simple group is isomorphic to one of the known ones. Another basic problem is the study of properties of the known simple groups: the study of matrix representations for them (cf. [[Finite group, representation of a|Finite group, representation of a]]); the description of all primitive permutation representations (cf. [[Permutation group|Permutation group]]) or, more generally, representations as automorphism groups of various mathematical objects (graphs, finite geometries); the description of the subgroups, in particular, maximal subgroups; etc.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.W. Carter,   "Simple groups of Lie type" , Wiley (Interscience)  (1972)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D. Gorenstein,   "Finite simple groups. An introduction to their classification" , Plenum  (1982)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B. Huppert,   "Endliche Gruppen" , '''1''' , Springer  (1967)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N. Blackburn,   B. Huppert,   "Finite groups" , '''2–3''' , Springer  (1984)</TD></TR></table>

====Comments====
Although, as of 1990, some parts of the full proof have not yet appeared in official journals, the classification of finite simple groups has been commonly accepted ever since 1982. The result is that, apart from those above, the only other finite simple (non-Abelian) groups are  $  2 $
sporadic simple groups, which together with the  $  5 $
Mathieu groups form the list of  $  26 $
groups given in [[Sporadic simple group|Sporadic simple group]].

The  "Atlas" , [[#References|[a1]]], is a good source for constructions, properties and references regarding these groups. See [[#References|[2]]] for an outline of the classification. See [[#References|[a2]]], [[#References|[a3]]] for information on the ongoing determination of maximal subgroups of the finite simple groups.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.H. Conway,   R.T. Curtis,   S.P. Norton,   R.A. Parker,   R.A. Wilson,   "Atlas of finite groups" , Clarendon Press  (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.B. Kleidman,   M.W. Liebeck,   "A survey of the maximal subgroups of the finite simple groups"  ''Geom. Dedicata'' , '''25'''  (1988)  pp. 375–389</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P.B. Kleidman,   M.W. Liebeck,   "The subgroup structure of the finite classical groups" , Cambridge Univ. Press  (1990)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D. Gorenstein,   "The classification of finite simple groups" , '''1. Groups of noncharacteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085210/s08521083.png" /> type''' , Plenum  (1983)</TD></TR></table>

{{SourceattributionEOM|Simple finite group}}
[[Category:Mathematics]]