I've been looking into profinite groups, their topological subgroup lattices, etc. I asked the question does every profinite group admit maximal subgroups?
I can't find an example of a profinite group which does not admit maximal subgroups, though I also cannot prove that each must have one. I think that if this is true, then it is related to profinite groups being residually finite, something about being able to tell apart elements using a projection onto a finite group, which must admit maximal subgroups.
Can someone answer this question with the affirmative, or give a counterexample of a profinite group which has no maximal subgroups?
Best Answer
If $G$ is any group and $H\subset G$ is a proper subgroup of finite index, then $H$ is contained in a maximal subgroup of $G$. Indeed, just take any proper subgroup containing $H$ of minimal index.
So in particular, any group with a proper subgroup of finite index (such as any nontrivial profinite group) has a maximal subgroup.