TITLE: Has the Jacobson/ Baer radical of a group been studied?
QUESTION [12 upvotes]: On groupprops, the Jacobson or Baer radical of a group $G$ is defined to be the intersection of all maximal normal subgroups of $G$. This is similar to, but distinct from, the Frattini subgroup which is the intersection of all maximal subgroups of $G$. 
Obviously the Jacobson radical of rings and the Frattini subgroup have been well-studied and are very useful.
Are there any significant results concerning or uses of the Jacobson radical of groups?
For comparison, significant results for the Frattini subgroup would include the facts about in on the Wikipedia page. So equivalent characterisations, its properties and so on.

REPLY [4 votes]: In addition to applications to finite/finitely generated groups, it's also useful in the study of locally compact groups (taking a definition that only allows closed normal subgroups).  Other names for it include the cosocle, the Mel'nikov subgroup or the normal Frattini subgroup.
In a connected Lie group $G$, $J(G)$ is the largest (not necessarily connected) soluble closed normal subgroup of $G$, and $G/J(G)$ is a direct product of simple Lie groups.  Given the Gleason-Yamabe theorem, similar statements apply more generally to connected locally compact groups.
If $G$ is a profinite group, then every proper closed normal subgroup is contained in a maximal open normal subgroup, and $G/J(G)$ is the largest quotient that is a Cartesian product of finite simple groups.  Moreover, the series $G > J(G) > J(J(G)) > \dots$ always has trivial intersection, so it can be a useful characteristic series, especially if the terms are known to have finite index (equivalently, if $J(H)$ has finite index for every open subgroup $H$).  Zalesskii showed that if $G$ is an infinite profinite group and $J(H)$ has finite index for every open subgroup $H$, then $G$ has a just infinite quotient.
If $G$ is a compactly generated locally compact group with no infinite discrete quotient, then Caprace--Monod showed that there is a smallest cocompact normal subgroup $N$ of $G$, and $N/J(N)$ is generated topologically by a finite set of topologically simple closed normal subgroups, with $N > J(N)$ unless $N$ is connected and soluble.