A definition of discrete group

Question

Definition: A discrete group $\Gamma$ is called residually finite if there exist subgroups $\Gamma\supset\Gamma_{1}\supset\Gamma_{2}\supset...$ such that each $\Gamma_{i}$ is a finite-index, normal subgroup of $\Gamma$ and $\bigcap_{n}\Gamma_{n}=\{e\}$(the neutral element of $\Gamma$).

I have two questions about this definition:

1. What is the so called "neutral element" of $\Gamma$?

2. Why is this group called "residually finite"? Is $\Gamma/\Gamma_{n}$ (quotient group) a finite group or somethingelse?

Answer 1

The terminology goes back to P. Hall. He called a group $G$ residually C, if for any element $g\neq 1$ in $G$ there exists a quotient group $G^*(g)$ belonging to C, such that the map $g^*\in G^*(g)$ is not the unit element in $G^*(g)$. Now put for $C$ just the class of finite groups to obtain residually finite groups.

Answer 2

The definition you wrote says taht $\Gamma_i$ is finite-index, which means precisely that $\Gamma/\Gamma_i$ is finite.