Hereditarily minimal topological groups

Speaker
Dr. Menachem Shlossberg, University of Udine
Date
31/12/2017 - 13:00 - 12:00Add to Calendar 2017-12-31 12:00:00 2017-12-31 13:00:00 Hereditarily minimal topological groups 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. Colloquium room, Mathematics Dept., building 216 אוניברסיטת בר-אילן - Department of Mathematics mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
Colloquium room, Mathematics Dept., building 216
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.

Last Updated Date : 25/12/2017