$|G| = pq$ with $p$, $q$ primes not necessarily distinct: case $p=q$
33300
Showing order of $Z (G) =1$ or $pq$
20881
Prove that if $p$ and $q$ are distinct primes such that $pq\mid n^2$ then $pq\mid n$.
136064
How can I prove that $\gcd(a,b)=1\implies \gcd(a^2,b^2)=1$ without using prime decomposition?
26449
$|G|=24$ prove that $G$ is not simple
47239
For $p,q$ primes such that $q|p-1$, show there is only one non-abelian group of order $pq$, up to isomorphism
45454
Pairs of $(p,q)$ such that $p^2 + 1 = 2 q^2$?
67272
Deduce that if $G$ is a finite $p$-group, the number of subgroups of $G$ that are not normal is divisible by $p$
20980
Prove that $\frac{p^p-1}{p-1}$ is not prime if $p \equiv 1 \pmod 4$
15526
Question on groups of order $pq$
95534
If $|G|=p^nq$, then $G$ contains a unique normal subgroup of index $q$
112191
How can I show $q^k$ does not divide ${{n}\choose{q}}$ when $q$ is a prime factor of $n$ and $k$ is the largest integer such that $q^k$ divides $n$?
43580
Let $p$ be a prime number and $G$ a non abelian group or order $p^3$. Prove that $Z(G) = [G,G]$.
88440
Is G a CLT group if its order is product of distinct prime?
51373
$G$ non abelian, order $p^3$ ($p$ prime). Suppose that the center is $p^2$, prove that $\exists\ x$ outside of the center, of order p
118629
$p,q$ odd primes, $q\mid a^p-1\Rightarrow q\mid a-1$ or $\,2p\mid q-1$
37183
Show that the set of all $x$ such that $x \in P$ and $x \notin Q$ exists
23885
Structure of a group, $G$, of order $pq$ where $p, q$ are prime.