Profinite group

A profinite group is an inverse limit of finite groups.

Let $I$ be a directed set respect to a relation $\leq$, where $\leq$ is reflexive and transitive and for every $i_1, i_2 \in I$, then there exists $i$ such that $i \geq i_1$ and $i \geq i_2$. An inverse system of topological spaces over $I$ is denoted $(I, G_i, \pi_i^j)$ where for each $i \in I$, $G_i$ is a topological space and for $i \leq j$, there exists a morphism $\pi_i^j :G_j \rightarrow G_i$ with $G_i^i$ being the identity. Also, if we have $i \leq j \leq k$, then then composition $\pi_i^j \pi^k_j = \pi^k_i$. Often we call it $(G_i)$.

Example: Consider $\mathbb{Z}/p^i \mathbb{Z}$ with usual $\leq$ of integers. The morphism $\pi_i^j$ for $i \leq j$ is the natural projection $\mathbb{Z}/p^j \mathbb{Z} \twoheadrightarrow \mathbb{Z}/p^i \mathbb{Z}$. Then it is easily followed that the composition is followed. 

We can view finite groups as a topological space by giving them the discrete topology (i.e. every subset of $G_i$ is an open set.) Let $(G_i)$ be an inverse system of finite group and consider $X = \prod G_i$, the cartesian product of all finite groups and make the group into a topological group by the following:

If $p_i : \prod G_i \rightarrow G_i$ are canonical projections, then the product topology on $X$ is the coarsets topology on $X$ such that all projections $p_i$ are continuous. Some call it the Tychonoff topology. The open sets in the product topology are the unions of the set of the form $\prod_{i \in I} U_i$ such that $U_i$ is open in $X_i$ and $U_i \neq X_i$ for only finitely many $i$.

Then if $L \subset \prod G_i$ be a subset of all elements of the form $(x_i)$ such that for all $i \leq j$, $\pi_i^j (x_j) = x_i$. Suppose $(x_i), (y_i) \in G$, then $(x_iy_i) \in G$ because for all $i \leq j$, $\pi_i^j(x_jy_j) = \pi_i^j(x_j) \pi_i^j(y_j) = x_iy_i$, hence $L$ becomes a subgroup of $\prod G_i$. Then we give $L$ the induced topology, and call it the inverse limit of the system $(G_i)$. If $(G_i)$ are finite $p$-groups, we call L the pro-$p$-group. Inverse limit is denoted,

$L = \varprojlim G_i$

Example: Continuing the example above, the inverse limit of $(\mathbb{Z}/p^i \mathbb{Z})$ is defined to be $\mathbb{Z}_p$, the $p$-adic integers. $\mathbb{Z}_p$ can be viewed as a power serises, i.e. every element is of the form $\sum_{i=0}^\infty a_ip^i$ where $a_i \in \{0, \ldots, p-1 \}$. Then arguing similarly with the fact that sequence of $\{ 0, 1 \}$ are countable, we see that $\mathbb{Z}_p$ is infinite. The quotient field of $\mathbb{Z}_p$ is denoted $\mathbb{Q}_p$ which has elements of the form $\sum_{i=k}^\infty a_ip^i$ with $a_i \in \{ 0, \ldots, p-1 \}$ for some integer $k$, and can be viewed as a completion of $\mathbb{Q}$ with the $p$-adic valuation.

Proposition: L is closed in $\prod G_i$.

Proof) If $(x_i) \notin L$, then there exists $i \leq j$ such that $\pi_i^j(x_j) \neq x_i$, then the following set $\prod_{k \in I, k \neq i, j} G_i \times \{x_i \} \times \{ x_j \}$ is open because $G_i$ are equipped with discrete topology. Also, the open set does not intersect with $L$, i.e. $(x_i)$ is not a limit point of $L$.