Hereditarily minimal topological groups

Sun, 31/12/2017 - 12:00
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Abstract: 

A Hausdorff topological group $(G, \tau)$ is called minimal if there exists no Hausdorff group topology on $G$ which is strictly coarser than $\tau$.

 

We say that a topological group $G$ is hereditarily  minimal, if every subgroup of $G$ is minimal.

 

By Prodanov's Theorem an infinite compact abelian group $K$ is isomorphic to $\Z_p$ (p-adic integers) for some prime $p$ if and only if  $K$ is hereditarily minimal.

 

We study hereditarily minimal groups. The following theorem is one of our main results.

 

Theorem

Let $G$ be an infinite hereditarily minimal locally compact group that is either compact or locally solvable. Then $G$ is either center-free or isomorphic to $\Z_p$, for some prime $p$.

 

In particular,

Corollary

If $G$ is an infinite hereditarily minimal locally compact nilpotent group, then $G$ is isomorphic to $\Z_p$ for some prime $p$.

 

This is a joint work with D. Dikranjan, D. Toller and W. Xi.