Let $G$ be a group and let $H$,$K$ be its subgroups such that $H \cap K=1$. From the hypothesis $H$ centralizes $K$, is it correct to conclude that $K=1$?
I'm reasoning as follows: if $H$ centralizes $K$, we have $H \leq C_G(K) \leq N_G(K)$, so $K \trianglelefteq H$. In particular, $K \leq H$, therefore $H \cap K= K$. Since $H \cap K=1$, we have $K=1$.
Is my thought correct? Thanks for the tips
$H\le N_G(K)$ doesn't necessarily imply that $K\le H$, let alone $K\trianglelefteq H$.
A counter-example to what you mentionned is a direct product $H\times K$; each factor in a direct product centralizes the other factor.
On the other hand, there are subgroups called malnormal. In your example, if $K$ is assumed to be malnormal, then it follows indeed that $K$ is trivial.