If G be a non-trivial group with no non-trivial proper subgroup, show that G is a group of prime order.
22301
If $G$ has no proper subgroup, then $G$ is cyclic of prime order
128300
Prove that a group with exactly two proper nontrivial subgroups is isomorphic to $\mathbb{Z}_{pq}$ or $\mathbb{Z}_{p^3}$.
134376
Let $G$ be a nonabelian group. Let $a \in G$ have order $4$ and let $b \in G$ have order $3$. Then the order of the element $ab$ in $G$
155054
$G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then $G=HK$?
2761
If $S$ is a nonempty subset of group $G$, then $S^{|G|}$ is a subgroup of $G$.
50762
If $S_n$ has cyclic subgroups of these orders or not?
157819
Every group of order $56$ has a proper non - trivial normal subgroup?
8841
A $p$-group of order $p^n$ has a normal subgroup of order $p^k$ for each $0\le k \le n$
134353
A group of order $108$ has a proper normal subgroup of order $\geq 6$.
88344
Order of a group and subsequent subgroups
61379
Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$.
67973
Prove that a finite abelian group is simple if and only if its order is prime.
61546
Prove that for each prime $p$ there exists a nonabelian group of order $p^3$
151727
If $|G/H|=4$ then $G$ is union of three proper subgroups
101735
A finite $p$-group cannot be simple unless it has order $p$
96878
A finite group with the property that all of its proper subgroups are abelian
44152
If $G$ has only one subgroup of order $n$, then that Subgroup is Normal