Sylow theorems

In mathematics, specifically in the field of <a href="/facts/Finite_group_theory/vv2tvogB">finite group theory</a>, the Sylow theorems are a collection of <a href="/facts/Theorem/f2Pe1FMD">theorems</a> named after the Norwegian mathematician <a href="/facts/Peter_Ludwig_Mejdell_Sylow/G046M90K">Peter Ludwig Sylow</a> that give detailed information about the number of <a href="/facts/Subgroup/r91mBgpa">subgroups</a> of fixed <a href="/facts/Order_of_a_group/Cl3tKGFg">order</a> that a given <a href="/facts/Finite_group/vv2tvogB">finite group</a> contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the <a href="/facts/Classification_of_finite_simple_groups/nMwLJf1K">classification of finite simple groups</a>.
For a <a href="/facts/Prime_number/L20FLkgn">prime number</a> 
 
 
 
 p
 
 
 {\displaystyle p}
 
, a <a href="/facts/P-group/MHeGQT9S">p-group</a> is a group whose <a href="/facts/Cardinality/m6VPojwT">cardinality</a> is a <a href="/facts/Power_(mathematics)/pac6QY2g">power</a> of 
 
 
 
 p
 ;
 
 
 {\displaystyle p;}
 
 or equivalently, the <a href="/facts/Order_of_a_group_element/Cl3tKGFg">order</a> of each group element is some power of 
 
 
 
 p
 
 
 {\displaystyle p}
 
. A Sylow p-subgroup (sometimes p-Sylow subgroup) of a finite group 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is a <a href="/facts/Maximal_subgroup/XebALFlp">maximal</a> 
 
 
 
 p
 
 
 {\displaystyle p}
 
-subgroup of 
 
 
 
 G
 
 
 {\displaystyle G}
 
, i.e., a subgroup of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 that is a p-group and is not a proper subgroup of any other 
 
 
 
 p
 
 
 {\displaystyle p}
 
-subgroup of 
 
 
 
 G
 
 
 {\displaystyle G}
 
. The set of all Sylow 
 
 
 
 p
 
 
 {\displaystyle p}
 
-subgroups for a given prime 
 
 
 
 p
 
 
 {\displaystyle p}
 
 is sometimes written 
 
 
 
 
 
 Syl
 
 
 p
 
 
 (
 G
 )
 
 
 {\displaystyle {\text{Syl}}_{p}(G)}
 
.
The Sylow theorems assert a partial converse to <a href="/facts/Lagrange%2527s_theorem_(group_theory)/VgyNjGlz">Lagrange's theorem</a>. Lagrange's theorem states that for any finite group 
 
 
 
 G
 
 
 {\displaystyle G}
 
 the order (number of elements) of every subgroup of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 divides the order of 
 
 
 
 G
 
 
 {\displaystyle G}
 
. The Sylow theorems state that for every <a href="/facts/Prime_factor/L20FLkgn">prime factor</a> 
 
 
 
 p
 
 
 {\displaystyle p}
 
 of the order of a finite group 
 
 
 
 G
 
 
 {\displaystyle G}
 
, there exists a Sylow 
 
 
 
 p
 
 
 {\displaystyle p}
 
-subgroup of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 of order 
 
 
 
 
 p
 
 n
 
 
 
 
 {\displaystyle p^{n}}
 
, the highest power of 
 
 
 
 p
 
 
 {\displaystyle p}
 
 that divides the order of 
 
 
 
 G
 
 
 {\displaystyle G}
 
. Moreover, every subgroup of order 
 
 
 
 
 p
 
 n
 
 
 
 
 {\displaystyle p^{n}}
 
 is a Sylow 
 
 
 
 p
 
 
 {\displaystyle p}
 
-subgroup of 
 
 
 
 G
 
 
 {\displaystyle G}
 
, and the Sylow 
 
 
 
 p
 
 
 {\displaystyle p}
 
-subgroups of a group (for a given prime 
 
 
 
 p
 
 
 {\displaystyle p}
 
) are <a href="/facts/Conjugacy_class/AFuPIFNW">conjugate</a> to each other. Furthermore, the number of Sylow 
 
 
 
 p
 
 
 {\displaystyle p}
 
-subgroups of a group for a given prime 
 
 
 
 p
 
 
 {\displaystyle p}
 
 is congruent to 1 (mod 
 
 
 
 p
 
 
 {\displaystyle p}
 
).

Sylow theorems open-in-new

Sylow theorems