# Normal covering number and maximal cyclic subgroups

For a finite non-cyclic group $G$ we say that a collection $\mathcal H$ of proper subgroups of $G$ is a cover if $G = \bigcup_{H \in \mathcal H} H$. If $H^g \in \mathcal H$ for any $H \in \mathcal H$ and $g \in G$, we say that $\mathcal H$ is a normal cover. One can now define $\gamma(G)$ to be the minimal number of conjugacy classes in a normal cover of $G$. This number has been investigated in a couple of papers. One could restrict the subgroups in the collections $\mathcal H$ to have additional properties, for example one could require them to be abelian or cyclic. In the latter case we would then count the number of conjugacy classes of maximal cyclic subgroups of $G$ (let us denote this number of $\gamma_c(G)$).

Has the quantity $\gamma_c(G)$ been investigated thoroughly in the literature? I have only found very little information.

• When you say "additional", I'm guessing you mean additional to being "proper subgroups", and not additional to being "normal proper subgroups"? Because while very group $G$ has a cyclic cover (and hence an abelian cover), not every group has a normal cover, let alone a normal abelian or normal cyclic cover... – Arturo Magidin Apr 4 '17 at 18:58
• Yes, I do not want to assume that the proper subgroups are normal. By a normal cover I do not mean that the subgroups of the cover are normal, only that the set of subgroups constituting the cover is closed under conjugation. – scalar Apr 4 '17 at 21:41
• Ah, sorry; I misinterpreted the term, since "abelian cover" is a cover by abelian groups, etc. – Arturo Magidin Apr 4 '17 at 21:44
• @scalar Please let me know what are the "very little information" that you have have found so far. – Alireza Abdollahi Apr 5 '17 at 3:59
• There is a paper by G.A. Miller from 1912 titled "Note on the maximal cyclic subgroups of a group of order $p^m$" that is related – scalar Apr 5 '17 at 4:35