What does it mean to have no proper non-trivial subgroup 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
 A: 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.
A: A subgroup N of a group $G$ is said to be proper if $N\neq G$ and to be non-trivial if $N\neq \{e\}$, where $e$ is the identity of $G$.
For example $N=\{0,2\}$ is a proper subgroup of $(\Bbb Z/4\Bbb Z,+)$, isomorphic to $\Bbb Z/2\Bbb Z$.
A: If a nontrivial group has no proper nontrivial subgroup, then it is a cyclic group of prime order. In other words, it is generated by a single element whose order is a prime number.
A: As you know 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.
Now a proper non trivial subgroup means it is neither identity nor G it. This is subgroup other than Identity and G itself.
