Contents

group theory

# Contents

## 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.

## References

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