Marcos Zyman, BMCC

Abstract:
Given a nilpotent group $G$ and a prime number $p$, there is a unique $p$-local group $G_{(p)}$ which is, in some sense, the ``best approximation" to $G$ among all $p$-local nilpotent groups. $G_{(p)}$ is called the $p$-localization of $G$. Let $IA(G)$ be the group of automorphisms of $G$ that induce the identity on $G/[G,G]$. $IA(G)$ turns out to be nilpotent so its $p$-localization exists.

Two nilpotent groups are said to be in the same localization genus if their $p$-localizations are isomorphic for all $p$. I will discuss the following theorem: if two finitely generated, torsion-free nilpotent, and metabelian groups lie in the same localization genus, their $IA$-groups also lie in the same localization genus. I also plan to present some background leading to this result, as well as a few examples.