  For a semi-stable abelian variety A_K over a complete discrete valuation
field K, we show that every finite subgroup scheme of A_K extends to a log
finite flat group scheme over the valuation ring of K endowed with the
canonical log structure. To achieve this, we first prove that every weak log
abelian variety over an fs log scheme with its underlying scheme locally
noetherian, is a sheaf for the Kummer flat topology, which answers a question
of Chikara Nakayama. We also give several equivalent conditions defining
isogenies of log abelian varieties.
