Finite groups $G$ such that $Soc(G)$ is abelian and contains normal subgroups only This question is a generalization of a previous one: Groups such that any nontrivial normal subgroup intersects the center nontrivially (see Jack Schmidt's answer):
What finite groups $G$ have the property that $Soc(G)$ is abelian and all of the subgroups of $Soc(G)$ are normal in $G$? (the linked question talks about the case where $Soc(G)$ is central and thus is a special case).
 A: The following are equivalent for an abelian subgroup N of the finite group G:


*

*every subgroup of N is normal in G

*every cyclic subgroup of N is normal in G

*for every g in G, there is a positive integer k, such that for every x in N, xg = xk.


In this case we say that G acts as power automorphisms on N.
When N = Soc(G), then we can add another condition:


*

*every minimal normal subgroup is cyclic of prime order, and those of the same order are isomorphic as modules


Suppose that not only G, but also H = G/Soc(G), and K = H/Soc(H), etc. all act as power automorphisms on their socle.  In this case, not only is every minimal normal subgroup cyclic, but in fact every chief factor is cyclic, and so the group is what is called supersolvable.  A finite nilpotent group is one in which every chief factor is the 1-dimensional trivial module.  A finite supersolvable group is one in which every chief factor is (any) 1-dimensional module.
Power automorphisms are discussed a bit in Roland Schmidt's textbook on the Subgroup Lattices of Groups.
When all of the p-chief factors for each prime p are required to be isomorphic inside a particular normal subgroup (not the socle), then I believe this characterizes "PST" groups.  In case we require it in the socle, I'm not immediately sure, but there might be something important there.
