Abstract :
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.
