$G_1$ is a group of order $p$ and $G_2$ a group of order $q$, where $p, q$ are primes. Prove that the direct product $G_1\times G_2$ is cyclic.
115726
For $p,q$ prime and $p \neq q$ show that $C_p \times C_q \cong C_{pq}$
83654
Order of an element in a finite cyclic group
8602
Galois Group of $(x^2-p_1)\cdots(x^2-p_n)$
99205
Group of order $p^k q$ where $p$ and $q$ are distinct primes - how to prove this property?
92430
Show that the group is cyclic.
15526
Question on groups of order $pq$
121810
Prove that the product of two infinite cyclic groups is not infinite cyclic.
107716
Show that $G$ is cyclic
42096
Direct product of Sylow subgroups
73402
Showing that $(\mathbb{Z}/p^{a}\mathbb{Z})^*$ is a cyclic group
14890
Let $G=\langle a\rangle$ be a cyclic group of order $n$. Show that, for every divisor $d$ of $n$, there exists a subgroup of $G$ whose order is $d$.
103423
Normal subgroup where G has order prime
3829
Number of automorphisms of a direct product of two cyclic $p$-groups
47478
Show that any cyclic group of even order has exactly one element of order $2$
98155
Free products of cyclic groups
89382
In a group $G$ where $\exists!$ nontrivial, proper subgroup, show that $G$ is cyclic and $\lvert G\rvert=p^2$, for $p$ prime.
23885
Structure of a group, $G$, of order $pq$ where $p, q$ are prime.