If $p$ divides $q-1$ where $p,q$ are primes and $p<q$ then how to show there exist a non abelian group of order $pq$?
49162
Classify all groups with order $pq$, $p$ is not equal to $q$
66877
Showing groups of order $p^{k}(p+1)$ are not simple, p prime
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 $n$?
134963
Proving a non-prime (composite) number $> 1$ has a divisor $d \leq\sqrt q$
100984
Prove that if prime $p$ divide $a_{2k}-2$, then $p$ divide also $a_{2k+1}-1$.
33300
Showing order of $Z (G) =1$ or $pq$
131706
Is my proof that $U_{pq}$ is not cyclic if $p$ and $q$ are distinct odd primes correct?
150188
Construct a non-abelian group of order 75
126893
A group of order 2p (p prime) and other conditions - prove abelian.
77806
Proving if $a$ is not a multiple of prime $p$, then an integer $b$ exists such that $p^b - 1$ is a multiple of $a$
88718
A group is generated by two elements of order $2$ is infinite and non-abelian
15341
Non-abelian finite group in which more than half of the elements have order $2$
122970
A question on $p$-groups, and order of its commutator subgroup.
142894
Showing that a group of order $pq$ is cyclic if it has normal subgroups of order $p$ and $q$
5841
Proving prime $p$ divides $\binom{p}{k}$ for $k\in\{1,\ldots,p-1\}$
97021
Do there exist two primes $p<q$ such that $p^n-1\mid q^n-1$ for infinitely many $n$?
158106
Prove that 1 less than the number of equivalence classes divides $p-1$ where $p$ is prime