If $G$ has no nontrivial subgroups, show that $G$ must be finite of prime order
53990
Group with more than one element and with no proper, nontrivial sub groups must have prime order.
51352
Subgroups of a finite abelian group
85579
$G$ a group s.t. every non-identity element has order 2. If $G$ is finite, prove $|G| = 2^n$ and $G \simeq C_2 \times C_2 \times\cdots\times C_2$
17178
Order of $\phi(g)$ divides the order of $g$
37913
Prove that $ AB$ is a subgroup of order $mn$ if $m$ and $n$ are relatively prime.
61379
Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$.
114535
what groups have only elements of prime order?
4402
Group of order $pqr$, $p < q < r$ primes
95534
If $|G|=p^nq$, then $G$ contains a unique normal subgroup of index $q$
9612
Frattini subgroup $Φ ( G )$ is not trivial
23885
Structure of a group, $G$, of order $pq$ where $p, q$ are prime.
7973
Groups of order $pq$ have a proper normal subgroup
43580
Let $p$ be a prime number and $G$ a non abelian group or order $p^3$. Prove that $Z(G) = [G,G]$.
14611
$A_4$ has no subgroup of order $6$?
158282
Finite order of an element $f(g)$ of a homomorphism $f$
129937
How to prove that $A_5$ has no subgroup of order 30?
19458
Finite groups and Normal subgroups - $G$ is a finite group such that $|G|=n$. Let $p$ be the smallest prime such that $p|n$.