Profinite groups

Profinite spaces

Usually, the prefix ‘’pro’’ is short for ‘’projective limit’’, which is another name of ‘’limit’’. In this sense, ‘’profinite’’ means ‘’limit of finite’’. Then, a profinite set should be a limit of finite sets. But in what category? One can take the limit in $\mathrm{Set}$, or in $\mathrm{Top}$ to get a topology on that set. So, we define
Definition: A profinite space is a limit of finite sets (viewed as discrete sets) in $\mathrm{Top}$.

But how the open sets in a profinite space look like?

Example: product of finite sets

Recall that the product topology on the Cartesian product $\prod_iX_i$ is given by the basis $$
\prod_i U_i,
$$ where $U_i$ are open sets of $X_i$ and only finitely many $U_i$ are not the entire $X_i$.

From this, one can see the topology of a product of finite sets must be quasi-compact, Hausdorff and totally disconnected.

Stone spaces

A Stone space is a topological space which is quasi-compact, Hausdorff and totally disconnected. Note that all those properties can be inherited by subspaces.

Recall that any limit is a subspace of a product. Then we get
Proposition: Profinite spaces are Stone spaces.

Conversely, let $X$ be a Stone space, we expect it to be a limit. So, first of all, we need a system of spaces under $X$, in particualr, a system of quotients of $X$. A possible quotient is the set of connected components. However, as we will see, this set is not always finite. Instead, we consider the set of closed-open partitions. A closed-open partition of $X$ is a collection of nonempty pairwise disjoint closed-open subsets of $X$ such that their union is $X$. The topology on a closed-open partition $I$ is the quotient topology induced by the projection $X\to I$. Since $X$ is quasi-compact, each $I$ must be finite. As quotients of $X$, there exist canonical projections between those closed-open partitions making them form a cofiltered system of quotients of $X$. Now, we have a canonical continuous map $$
X\longrightarrow \varprojlim I.
$$ This map is injective since the totally disconnectedness implies that any two distinct points can be separated by at least a pair of closed-open sets. This map is also surjective since the compatibility of members of the system implies that for any $(U_I)\in \varprojlim I$, each the intersection of any two $U_I$ is nonempty and then the quasi-compactness implies that the intersection of all $U_I$ is nonempty and hence gives the preimage of $(U_I)$. Finally, this map lies in the category of quasi-compact Hausdorff spaces. As this category is balanced, we conclude that the map is a homeomorphism.

In this way, we see:
Proposition: Stone spaces are profinite spaces.

Profinite groups

Likewise, a profinite group is a limit of finite groups in the category of topological groups. Recall the forgetful functors to $\mathrm{Top}$ and $\mathrm{Grp}$ preserve limits, therefore a profinite group is precisely a topological group whose underlying topological space is a profinite space.

Open subgroups

Let $G$ be a profinite group. First, consider the following question: what is the intersection of all closed-open neighborhoods of $1$? Of course it should contain the connected component of $1$ which is ${1}$. Then, the compactness implies that the intersection of all closed-open neighborhoods of $1$ is connected. Therefore, this intersection must be ${1}$.

Next, what’s a neighborhood basis of $1$? An obvious one is the system of all closed-open neighborhoods of $1$. The continuousness of operations further implies that the subsystem of all closed-open subgroups of $G$ is also a neighborhood basis of $1$.

One may noticed that any open subgroup $U$ of $G$ must be also closed since its complement is the union of the cosets of $U$ which are not $U$. Moreover, since $G$ is quasi-compact, there must be only finitely many cosets. Recall that for any subgroup, the intersection of all its conjugates is a normal subgroup. We see that any open subgroup of $G$ contains an open normal subgroup.

Note that for any open norm subgroup $U$ of $G$, $G/U$ is a finite quotient group of $G$. Thus we have a canonical homomoprhism $$
G\longrightarrow\varprojlim G/U.
$$ This homomorphism is injective. As $G$ is Hausdorff, any two elements of $G$ can be separated by two open sets. Since open normal subgroups form a neighborhood basis of $1$, we can shrink the two open sets to two cosets of an open normal subgroup $U$ of $G$. This homomorphism is also surjective by the quasi-compactness. As we are dealing with quasi-compact Hausdorff groups, which form a balanced category, we conclude that the above homomorphism is an isomorphism.

Morphisms of profintie groups

Let $G, H$ be two profinite groups. Let’s consider what’s $\mathrm{Hom}(G,H)$?

First, we write $G$ and $H$ as $\varprojlim G/U$ and $\varprojlim H/V$ respectively. Then $$\mathrm{Hom}(G,H) = \mathrm{Hom}(\varprojlim_U G/U,\varprojlim_V H/V) = \varprojlim_V\mathrm{Hom}(\varprojlim_U G/U,H/V).$$

Since each $H/V$ is finite, we conclude that $$\mathrm{Hom}(G,H) = \varprojlim_V\varinjlim_U\mathrm{Hom}(G/U,H/V).$$

The above formula can help us to reduce problems from profinite groups to finite groups. For instance, recall that in the category of finite groups, monomorphisms (resp. epimorphisms) are injective (resp. surjective) homomorphisms. Thus the same statements hold for profinite groups.

Indices of closed subgroups

Let $G$ be a profinite group and $H$ a closed subgroup of it. Recall that the index $(G:H)$ is defined to be the least common multiple of $(G:U)$ where $U$ goes over the family of open subgroups of $G$ containing $H$. In particular, $\#G:=(G:1)$ is called the order of $G$.

The follwoing results follow from the defiinions and direct computations.

  • Let $H,K$ be closed subgroups of $G$, hence also profinite. If $K<H$, then $(G:K) = (G:H)(H:K)$. If moreover, $K$ is normal in $G$, then $(G/K:H/K) = (G:H)$.
  • A closed subgroup $H$ in $G$ is open if and only if $(G:H)$ is finite.
  • If ${H_i}$ is a decreasing filtered family of closed subgroups in $G$ and $H = \bigcap_iH_i$, then $(G:H)$ is the least common multiple of $(G:H_i)$.
  • If $G$ is a limit of profinite groups $G_i$ such that each transitaion map is surjective, then $\#G$ is the least common multiple of $\#G_i$.

Pro-$p$-groups

Let $p$ be a prime number. Like profinite groups, a pro-$p$ group is a limit of finite $p$ groups. Note that this is equivalent to say $\#G$ is a power of $p$. Then, for a profinite group $G$, the maximal pro-$p$ subgroups are called the Sylow $p$ subgroups. In more common setting, a Sylow $p$ subgroup is defined to be a closed subgroup $P$ of $G$ which is a pro-$p$ group and $p\nmid(G:P)$. The usual Sylow Theorems hold in this setting saying the equivalence of the above two definitions and that any two Sylow $p$ subgroups are conjugate.

Reference

For the notion of profinite groups and supernatural numbers, see

  • Michael D. Fried, Moshe Jarden, Field Arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete 3, Folge 11 (3rd ed.), Springer-Verlag, 2008.