TITLE: Almost normal subgroups: Is there any notion which is weaker than normal subgroup?
QUESTION [9 upvotes]: Let $G$ be a group then $N$ a subgroup of $G$ is said to be normal subgroup of $G$ if $\forall g \in G$, $g^{-1}Ng = N$. 
Is there any notion which is weaker than normal subgroup? I mean something like almost normal subgroup or nearly normal subgroup in which some of the elements only follow the normality condition.

REPLY [2 votes]: $N$ is said to be a commensurated subgroup if $[N:N\cap gNg^{-1}]$ is finite for every $g\in G$. This has many other names (almost normal, $(G,N)$ is a Hecke pair, etc). This notion occurs in various contexts.