Suppose a group G has no proper subgroups (that is, the only subgroup of G is G itself and the trivial subgroup {e}. Show that G is cyclic. Suppose a group G has no proper subgroups (that is, the only subgroup of G is G itself and the trivial subgroup {e}. Show that G is cyclic.

how to solve the question?
Hint

If $G=\{e\}$ then we are done. Now assume that there's $G\ni g\ne e$. What's the subgroup of $G$ generated by $g$?
if you Let G be a group with no proper, nontrivial subgroups.let g be a non
-trivial element of G then  Consider &lt; g >,the cyclic subgroup generated by g.
This is a subgroup of G, but since the only subgroups of G are &lt; e >and
G, &lt; g >=&lt; e >or &lt; g >= G. But we assumed g was nontrivial.
So&lt; g > not equal&lt; e > :Thus, &lt; g >= G:But this means G is generated by
&lt; g >