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)=\lbrace y \in G \mid \langle x‎, ‎y\rangle \; {\rm is \; cyclic} \rbrace‎. ‎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 ۱ \rbrace‎.

Keywords:

Finite groups , cyclicizers , tidy groups , C-tidy groups