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 su...