If a group has no subgroups other than the identity and itself, then it is finite and is of prime order
39976
If a group G has only the trivial group and G itself as its subgroup, can we be sure that G has prime order?
23885
Structure of a group, $G$, of order $pq$ where $p, q$ are prime.
47239
For $p,q$ primes such that $q|p-1$, show there is only one non-abelian group of order $pq$, up to isomorphism
88344
Order of a group and subsequent subgroups
145127
For any integers $m,n>1$ , does there exist a group $G$ with elements $a,b \in G$ such that $o(a)=m , o(b)=n$ but $ab$ has infinite order ?
39938
Subgroups of a cyclic group and their order.
96878
A finite group with the property that all of its proper subgroups are abelian
85106
Group of order 10 has an element of order 5, without using Cauchy's or Sylow's theorems
63499
Groups of prime squared order
96614
Finite number of subgroups $\Rightarrow$  finite group
129830
A group with no proper non-trivial subgroups
85579
$G$ a group s.t. every non-identity element has order 2. If $G$ is finite, prove $|G| = 2^n$ and $G \simeq C_2 \times C_2 \times\cdots\times C_2$
118849
If a cyclic group has an element of infinite order, how many elements of finite order does it have?
121423
Prove that every element of a finite group has an order
99205
Group of order $p^k q$ where $p$ and $q$ are distinct primes - how to prove this property?
145254
$G$ be a non-nilpotent and supersoluble finite group.
12667
How many subgroups of order $n$ does $D_n$ have?