I can't fully understand the definition of finite abelian section rank of a group.
By definition, a group $G$ has finite abelian section rank if it has no infinite elementary abelian $p$-sections for any prime $p$.
Can someone clarify what we mean by infinite elementary abelian $p$-sections?
A section of a groups $G$ is a quotient group $M/N$, where $M$ and $N$ are subgroups of $G$, and $N$ is a normal subgroup of $M$. A group is an elementary abelian $p$-group if it is abelian of exponent $p$. Presumably this means that the group has no sections that are infinite elementary abelian $p$-groups.