Central graphs and intersections of good subnormal subgroups

A subnormal subgroup $H$ of a group $G$ is called good if $\\langle H,S\\rangle$ is subnormal in $G$ for every subnormal subgroup $S$ of $G$. We construct a countable group $G$ and good subgroups $H,K\\le G$ for which $H\\cap K$ is not good, giving a negative answer to Kourovka Problem 8.74. Commuting square-zero endomorphisms of an $\\mathbb F_2$-vector space provide two subnormal subgroups with a non-subnormal join. A central graph construction then gives the required good subgroups and an explicit witness to the failure of goodness of their intersection.