Does every infinite group have a maximal subgroup? 
$G$ is an infinite group.

*

*Is it necessary true that there exists a subgroup $H$ of $G$ and $H$ is maximal ?


*Is it possible that there exists such series $H_1 < H_2 < H_3 <\cdots <G $ with the property that for every $H_i$ there exists $H_{i+1}$ such that $H_i < H_{i+1}$?


Rotman p. 324 problem 10.25:

The following conditions on an abelian group are equivalent:

*

*$G$ is divisible.


*Every nonzero quotient of $G$ is infinite; and


*$G$ has no maximal subgroups.

It is easy to see above points are equivalent. If you need the details, I can add them here.
