The p-section of a p-element

Let $G$ be a finite group, let $p$ be prime, and let $u\in G$ be a $p$-element. The $p$-section of $u$ is

$S_G(u):=\{g\in G:g_p\text{ is }G\text{-conjugate to }u\},$

where $g_p$ is the $p$-part from Every finite-group element has unique commuting p- and p-prime parts. Equivalently, $S_G(u)$ is the union of the $G$-conjugacy classes that meet

$\{uv:v\in C_G(u)\text{ is }p\text{-regular}\}\mathrm{.}$

Indeed, if $h{g}_p{h}^{-1}=u$, uniqueness and conjugation equivariance of the commuting parts give $hg{h}^{-1}=u(h{g}_{p^{\mathrm{\prime }}}{h}^{-1})$, where the second factor is $p$-regular and centralizes $u$. Conversely, the unique $p$-part of $uv$ is $u$ whenever $v$ is $p$-regular and centralizes $u$.

The section depends only on the conjugacy class of $u$. In particular, $S_G(1)$ is exactly the set of $p$-regular elements of $G$.