For $p,q$ prime and $p \neq q$ show that $C_p \times C_q \cong C_{pq}$
28362
Product of two cyclic groups is cyclic iff their orders are co-prime
96062
Show $\underbrace{{111\cdots}1}_{{\small{p-1} \ 1's}}$ is divisible by $p$
105139
Verification of Proof that a nonabelian group G of order pq where p and q are primes has a trivial center
60305
Proving the congruence $p^{q-1}+q^{p-1} \equiv 1 \pmod{pq}$
53266
Show that $R/I\otimes_R R/J\cong R/(I+J)$
99205
Group of order $p^k q$ where $p$ and $q$ are distinct primes - how to prove this property?
61103
Prove that $\mathbb{|Q| = |Q\times Q|}$
37093
If p is an odd prime, prove that $1^2 \times 3^2 \times 5^2 \cdots  \times (p-2)^2 \equiv (-1)^{(p+1)/2}\pmod{p}$
21250
Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$
79995
G is group of order pq, pq are primes
152481
If order of group is $p^2$, where $p$ is prime, how can you deduce $G$ is isomorphic to $C_{p^2}$ or $C_p \times C_p$?
27170
$a\in \mathbb{N}$, $p$ prime, $a<p$ prove that $a\mid p+1\iff\exists\, b,c\in\Bbb N:\dfrac{a}{p}=\dfrac{1}{b}+\dfrac{1}{c}$
72144
Show that $\mathbb Q(\sqrt p) \not\simeq\mathbb Q(\sqrt q)$
15276
$G \times H \cong G \times K$ , then $ K \cong H$
141683
$ |G_1 |$ and $|G_2 | $ are coprime. Show that $K = H_1 \times H_2$
73238
Show that $a^{q-1} \equiv 1 \pmod{pq}$
60804
About the property of $m$: if $n < m$ is co-prime to $m$, then $n$ is prime