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Sylow theorems
Theorems that help decompose a finite group based on prime factors of its order / From Wikipedia, the free encyclopedia
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In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow[1] that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups.
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For a prime number , a Sylow p-subgroup (sometimes p-Sylow subgroup) of a group
is a maximal
-subgroup of
, i.e., a subgroup of
that is a p-group (meaning its cardinality is a power of
or equivalently, the order of every group element is a power of
) that is not a proper subgroup of any other
-subgroup of
. The set of all Sylow
-subgroups for a given prime
is sometimes written
.
The Sylow theorems assert a partial converse to Lagrange's theorem. Lagrange's theorem states that for any finite group the order (number of elements) of every subgroup of
divides the order of
. The Sylow theorems state that for every prime factor
of the order of a finite group
, there exists a Sylow
-subgroup of
of order
, the highest power of
that divides the order of
. Moreover, every subgroup of order
is a Sylow
-subgroup of
, and the Sylow
-subgroups of a group (for a given prime
) are conjugate to each other. Furthermore, the number of Sylow
-subgroups of a group for a given prime
is congruent to 1 (mod
).