n-cyclicizer groups

‎The cyclicizer of an element $x$ of a group $G$ is defined as‎ ‎$Cyc_G(x)=\{y\in G|\big < x,y\big > $ is‎ ~‎cyclic$\}.$ Here‎, ‎we introduce an $n$-cyclicizer group and‎ ‎show that there is no finite $n$-cyclicizer group for $n=2,3$‎. ‎We‎ ‎prove that for any positive integer $n\neq 2,3$‎, ‎there exists a‎ ‎finite $n$-cyclicizer group and determine the structure of finite‎ ‎4 and 6-cyclicizer groups‎. ‎Also‎, ‎we characterize finite $5,7$ and‎ ‎$8$-cyclicizer groups‎.