# Almost normal subgroups: Is there any notion which is weaker than normal subgroup?

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.

A very important notion is that of subnormality: a subgroup $$H$$ of $$G$$ is called subnormal if there is a series $$H=H_0 \lhd H_1 \lhd \cdots \lhd H_{n-1} \lhd H_n=G$$. Of course every normal subgroup is subnormal, but the converse is not true in general. Helmut Wielandt basically established the theory in 1939, one of his famous results being the Zipper Lemma.

Other notions to be considered are pronormal subgroups and quasinormal or permutable subgroups, each of which generalizes a certain property of normal subgroups.

Consider the set of all conjugates of a subgroup $$H\leq G$$, defined by $$C=\{gHg^{-1}\mid g\in G\}$$. If $$H$$ is normal, clearly $$|C|=1$$, so $$|C|$$ can be considered to be a measure of "how far away from being normal" a subgroup is.

What about $$N$$ admits only a finite number oof conjugates ? For instance, when the group $$G$$ acts on a finite space, the stabilizer of a point has this property.

• The second sentence characterizes finite index subgroups. – YCor Feb 22 at 10:14

In the contex of locally compact topological groups which are equiped with the Haar measure, one can consider the following: "For almost all $$g\in G$$ we have $$g^{-1} Ng=N$$"

In the context of Lie groups, one can consider $$g^{-1} N g$$ would be isometric to $$N$$ with metric they inherit from a left (but not right) invariant metric.

In the context of abstract groups one can says "$$\cap_g g^{-1} N g$$ has finite index in $$N$$.

$$N$$ 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.