Definition 1

An increasing (resp. decreasing) sequence $(G_n)_{n \in \mathbf{Z}}$ of subgroups of a group $G$ is called an increasing (resp. decreasing) filtration on $G$. A group with a filtration is called a filtered group.

If $(G_n)_{n \in \mathbf{Z}}$ is an increasing (resp. decreasing) filtration on a group $G$ and we write $G'_n = G_{-n}$, clearly $(G'_n)_{n \in \mathbf{Z}}$ is a decreasing (resp. increasing) filtration on $G$. We may therefore restrict our study to decreasing filtrations and hence forth when we speak of a filtration, we shall mean a decreasing filtration, unless otherwise stated.

Given a decreasing filtration $(G_n)_{n \in \mathbf{Z}}$ on a group $G$, clearly $\bigcap_{n \in \mathbf{Z}} G_n$ and $\bigcup_{n \in \mathbf{Z}} G_n$ are subgroups of $G$; the filtration is called separated if $\bigcup_{n \in \mathbf{Z}} G_n$ is reduced to the identity element and exhaustive if $\bigcup_{n \in \mathbf{Z}} G_n = G$.