I am reading a first course in abstract algebra and there is a claim that says a group $G$ with no proper non trivial subgroups is cyclic. But I don't understand what does it mean to have no proper non trivial subgroup. I know that the identity element is trivial subgroup, all other subgroups are nontrivial and $G$ is the improper subgroup of $G$, and all others are proper subgroups. But what is proper non trivial subgroup? Thanks
[Math] What does it mean to have no proper non-trivial subgroup
abstract-algebracyclic-groupsdefinitiongroup-theory
Best Answer
For example, in $\{0,1,2,3\}$ (cyclic group of order $4$) the elements $\{0,2\}$ make a subgroup. This is a nontrivial subgroup, and it is not the entire group, so it is a proper subgroup.
The point is that a subgroup is also a subset. Subsets can be proper, or improper (i.e. equal to the big set). Proper subgroups are proper subsets.