Let $F$ be a family of finite groups closed under taking subgroups, quotients and finite direct products. Given an element $g$ of a profinite group $G$, the $F$-izer of $g$ in $G$ is the set of elements $x$ in $G$ such that $\langle g,x\rangle$ is a pro-$F$-group. Let $F(G)$ be the set of elements $g$ of $G$ such that the $F$-izer of $g$ in $G$ has positive Haar measure. In order to understand the set $F(G)$ and its influence on the structure of $G$, it is crucial to understand whether it is closed in $G$, whether it is a subgroup of $G$ and whether it admits a purely group-theoretical characterization. The group $G$ is called $F$-positive if $F(G)=G$. Detomi, Lucchini, Morigi and Shumyatsky proved that if $F$ is the class of finite solvable groups, then $F$-positivity is equivalent to being virtually prosolvable, and if $F$ is the class of finite nilpotent groups, then $F$-positivity is equivalent to being finite by pronilpotent. In this talk, we discuss generalizations of these results by relaxing the assumption that $F(G)=G$. Among other things, we prove that if $F(G)$ has positive Haar measure then $G$ is virtually pro-$F$ when $F$ is the class of finite solvable groups or finite $p$-groups (for a fixed prime $p$). We also prove that if an element $g$ of $G$ has nilpotentizer of positive Haar measure, then $g$ has finite order modulo the Fitting subgroup of $G$.
This is a joint work with Andrea Lucchini and Nowras Otmen.

There will be a coffe break after the seminar.