nLab Sylow p-subgroup

Definition

In a finite group $G$, and for a prime $p$, a maximal $p$-torsion subgroup of $G$ is also known as a Sylow $p$-subgroup.

Properties

The following statements are known as Sylow's theorems, a partial converse to Lagrange's theorem.

Let the order of $G$ be $r p^k$, where $r$ is coprime to $p$.

Let $H \lt G$ be a subgroup of rank $p^l$, and consider the left action of $G$ on right cosets of $H$:

$$(g, x H ) \;\mapsto\; (g x) H \,.$$

This induces a further action of $G$ on $(G / H) \mathbf{C} (p^{k-l})$, the subsets of $G/H$ of size $p^{k-l}$. The number of these, $\binom{ r p^{k-l} }{ p^{k-l} }$, is congruent to $r$, mod $p$, and hence some orbit must have size coprime to $p$, hence necessarily dividing $r$, hence some set of cosets must have stabilizer of size at least $p^k$. One checks that, on the other hand, the stabilizer of a set of cosets is at most the size of their union for a very good reason, and furthermore is a subgroup of $G$. Lastly, every orbit contains a representative that contains $H$. In consequence:

Theorem

Every $p$-subgroup $H$ of $G$ is contained in a subgroup of order $p^k$, which is necessarily a maximal $p$-subgroup. The number of maximal $p$-subgroups including $H$ is congruent to $1$ mod $p$.

One also has:

Theorem

Any two Sylow $p$-subgroups of $G$ are conjugate.

See class equation for a detailed discussion of these matters. (Now updated to take into account the proof below. The discussion above refers to a more involved proof from an earlier page version, which in turn was adapted from the Wikipedia article; it may be found here, comment 4.) The following slick proof for the existence of Sylow subgroups was suggested to us by Benjamin Steinberg.

Proof that Sylow subgroups exist

First observe that if a group $G$ has a $p$-Sylow subgroup $P$, then so does each of its subgroups $H$. For we let $H$ act on $G/P$ by left translation, and then note that since $G/P$ has cardinality prime to $p$, so must one of its connected components $H/Stab(a_x)$ in the $H$-set decomposition

$$G/P \cong \sum_{orbits\; x} H/Stab(a_x)$$

($a_x \in G/P$ a representative of its orbit $x$), making $Stab(a_x)$ a $p$-Sylow subgroup of $H$.

Then, if $H$ is any group, apply this observation to the embedding

$$H \stackrel{Cayley}{\hookrightarrow} Perm({|H|}) \hookrightarrow GL_{{|H|}}(\mathbb{Z}/(p)) \,,$$

where we embed the permutation group via permutation matrices into the group $G$ consisting of matrices ${|H|} \times {|H|} \to \mathbb{Z}/(p)$. Letting $n$ be the cardinality of $H$, the order of $G$ is $(p^n - 1)(p^n - p)\ldots (p^n - p^{n-1})$, with maximal $p$-factor $p^{n(n-1)/2}$. This $G$ has a $p$-Sylow subgroup given by unitriangular matrices, i.e., upper-triangular matrices with all $1$‘s on the diagonal, and we are done.

Related pages

• Sylow's theorem

• group actions on n-spheres

References

See also

• Wikipedia, Sylow theorems

For a generalisation of Sylow theory to $\pi$-finite ∞-groups, that is, ∞-groups with finitely many non-trivial homotopy groups which are all finite, see

• Matan Prasma, Tomer Schlank, Sylow theorems for ∞-groups (arXiv:1602.04494)