Near abelian profinite groups

A compact $p$-group $G$ ($p$ prime) is called {\it near abelian}, if it contains an abelian normal subgroup $A$ such that $G/A$ has a dense cyclic subgroup and that every closed subgroup of $A$ is normal in $G$. A detailed description of compact near abelian $p$-groups is given. Profinite near abelian groups turn out to be metabelian pronilpotent and so they are direct products of the $p$-primary components. Therefore the main results specialize at once to the consideration of compact $p$-groups. We relate near abelian groups to a class of compact groups, which are rich in permuting subgroups. A compact group is called {\it quasihamiltonian} (or {\it modular}), if every pair of compact subgroups commutes setwise. We show that for $p\ne 2$ a compact $p$-group $G$ is near abelian if and only if it is quasihamiltonian. The case $p=2$ is discussed separately.