Sylow Theorems
The Sylow theorems are a collection of results in the theory of finite groups. They give a partial converse to Lagrange's Theorem, and are one of the most important results in the field. They are named for P. Ludwig Sylow, who published their proof in 1872.
The Theorems
Throughout this article, will be a prime.
First, we show a lemma.
Lemma. Let , where
is an nonnegative integer and
is a nonnegative integer not divisible by
. Then
Proof. Let be a group of order
(e.g.,
, and let
be a set of size
. Let
act on the set
by the law
; extend this action canonically to the subsets of
of size
. Evidently, there are
such subsets.
Evidently, a subset of is stable under this action if and only if
. Thus the fixed points of the action are exactly the subsets of the form
, for
. Then there are
fixed points. Then
since the
-group
operates on a set of size
with
fixed points.
Let be a finite group.
Theorem 1. For every prime , every finite group contains a Sylow
-subgroup.
Proof. Let be a finite group of order
, for some positive integer
, not divisible by
. Let
denote the set of subsets of
of size
. Consider the action of
by left translation on the elements of
. There are
such subsets. Since
some orbit
of
must have cardinality not divisible by
. Since
, it follows that
; in particular,
. Since each element of
must be contained in one of the elements of
, it follows that the elements of
must be disjoint.
Consider now the equivalence relation on elements of
, defined as "
and
are in the same element of
". Then
is compatible with left translation by
; since the elements of
are disjoint,
is an equivalence relation. Thus the equivalence class of the identity is a subgroup
of
, which must have order
.
Theorem 2. The Sylow -subgroups of
are conjugates.
Theorem 3. The number of Sylow -subgroups of
is equivalent to 1 (mod
).