A group of order $p^2$, where $p$ is prime, with exactly one proper subgroup is cyclic.
89382
In a group $G$ where $\exists!$ nontrivial, proper subgroup, show that $G$ is cyclic and $\lvert G\rvert=p^2$, for $p$ prime.
116080
Let G be a finite group with more than one element. Show that G has an element of prime order
134820
$p$ prime, Group of order $p^n$ is cyclic iff it is an abelian group having a unique subgroup of order $p$
101602
consider the group $G=\mathbb Q/\mathbb Z$. For $n>0$, is there a cyclic subgroup of order n
79995
G is group of order pq, pq are primes
50706
$|G|=p(p+1)$ for $p$ prime, then $G$ has a normal subgroup of order $p$ or $p+1$
96878
A finite group with the property that all of its proper subgroups are abelian
54082
Group with exactly two subgroups of index 2
73211
Cyclic subgroups dealing with prime numbers Question
114535
what groups have only elements of prime order?
108309
An example of a non-abelian group of order $p^3$
9232
Group with proper subgroups infinite cyclic
142894
Showing that a group of order $pq$ is cyclic if it has normal subgroups of order $p$ and $q$
90449
Showing that a Sylow subgroup of a group of order $p^{2}q$ is normal.
54195
Do groups of prime order have one subgroup?
13734
Show that every group of prime order is cyclic
70001
Any group of order $2p$ has a subgroup of order $p~$($p$ being a prime)