$p$-Groups with Chernikov Centralizers of Non-Identity Elements of Prime Order

Let $G$ be a $p$-group, $a$ its element of prime order $p$, and $C_G(a)$ a Chernikov group. We prove that either $G$ is a Chernikov group, or $G$ possesses a non-locally finite section w. r. t. a Chernikov subgroup in which a maximal locally finite subgroup containing an image of $a$ is unique. Moreover, it is shown that the set of groups which satisfy the first part of the alternative is countable, while the set of groups which comply with the second is of the power of the continuum for every odd $p$.