A note on finite C-tidy groups

Let $G$ be a group and $x in G$. The cyclicizer of $x$ is defined to be the subset $Cyc(x)={ y in G | is cyclic}. $G$ is said to be a tidy group if $Cyc(x)$ is a subgroup for all $x in G$. We call $G$ to be a C-tidy group if $Cyc(x)$ is a cyclic subgroup for all $x in G setminus K(G)$, where $K(G)$ is the intersection of all the cyclicizers in $G$. In this note, we classify finite C-tidy groups with $K(G)=lbrace 1 rbrace$.