Dear All,
if $G$ is a group and $\langle a\rangle$ -- any its cyclic subgroup, is it true that there always exists a proper subgroup $H$ in $G$ with $G=\langle a\rangle H$? If "no", would it still be true for finite groups $G$?
Thank you!
P.S.: motivation comes from this -- one my colleague applied mathematician asked me if that would be true -- it somehow appears in his research, and I cannot see any counter-example to this.
One way to see many counterxamples for finite groups is to note that if $G$ has even order, then $G$ contains an element $a$ of order $2$. Any proper subgroup $H$ of $G$ with $G = \langle a \rangle H$ would have to be normal, since subgroups of index $2$ are always normal. Hence every non-trivial perfect finite group $G$ (of even order) provides a counterexample. In particular, every non-Abelian finite simple group (of even order) does. The parentheses are because every non-trivial perfect finite group does in fact have even order, but even without knowing that, there are plenty of explicit examples.
