  We say a group is finitely annihilated if it is the set-theoretic union of
all its proper normal finite index subgroups. We investigate this new property,
and observe that it is independent of several other well known group
properties. For finitely generated groups, we show that in many cases it is
equivalent to having non-cyclic abelianisation, and at the same time construct
an explicit infinite family of counterexamples to this. We show for finitely
presented groups that this property is neither Markov nor co-Markov. In the
context of our work we show that the weight of a non-perfect finite group, or a
non-perfect finitely generated solvable group, is the same as the weight of its
abelianisation. We generalise a theorem of Brodie-Chamberlain-Kappe on finite
coverings of groups, and finish with some generalisations and variations of our
new definition.
