Finitely annihilated groups

TL;DR

This paper introduces and explores finitely annihilated groups, a new class characterized by their union of proper normal finite index subgroups, revealing their properties, examples, and relations to other group properties.

Contribution

It defines finitely annihilated groups, investigates their properties, and establishes their independence from known properties, including explicit examples and theoretical generalizations.

Findings

Finitely annihilated groups are independent of several known properties.

For finitely generated groups, this property often aligns with having non-cyclic abelianisation.

Constructed infinite families of counterexamples to the equivalence.

Abstract

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.