query: $|G| = pqr$ with $p$, $q$ and $r$ distinct primes. Show G is not simple.
positive_passages.docid: 19016
positive_passages.text: Given 3 distinct primes {$p,q,r$}, then $|G|=pqr \implies G$ not simple
negative_passages.docid: 95111
negative_passages.text: Elementary proof for $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ where $p_i$ are different prime numbers.
negative_passages.docid: 6332
negative_passages.text: If $p\equiv1\mod{4}$, then $p=a^{2}+b^{2}$ where a and b are uniquely determined
negative_passages.docid: 44019
negative_passages.text: Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $.
negative_passages.docid: 103496
negative_passages.text: Show that $3p^2=q^2$ implies $3|p$ and $3|q$
negative_passages.docid: 131551
negative_passages.text: $pq\mid pn^q+qn^p+kn$ for some $k,\,$ if $p,q$ are primes
negative_passages.docid: 11564
negative_passages.text: $\mathbb{Q}(\sqrt{p},\sqrt[3]{q})=\mathbb{Q}(\sqrt{p}\sqrt[3]{q})$ for distinct prime $p,q$
negative_passages.docid: 105139
negative_passages.text: Verification of Proof that a nonabelian group G of order pq where p and q are primes has a trivial center
negative_passages.docid: 67973
negative_passages.text: Prove that a finite abelian group is simple if and only if its order is prime.
negative_passages.docid: 96062
negative_passages.text: Show $\underbrace{{111\cdots}1}_{{\small{p-1} \ 1's}}$ is divisible by $p$
negative_passages.docid: 59391
negative_passages.text: $p,q,r$ primes, $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is irrational.
negative_passages.docid: 129061
negative_passages.text: How to say that any group of order $pqr$ is cyclic with a provided relation between $p$, $q$, $r$, where they are all distinct primes.
negative_passages.docid: 60305
negative_passages.text: Proving the congruence $p^{q-1}+q^{p-1} \equiv 1 \pmod{pq}$
negative_passages.docid: 12066
negative_passages.text: Group of order pq is not simple
negative_passages.docid: 44729
negative_passages.text: Prove $g$ is a generator if $g^q=1 \pmod p$
negative_passages.docid: 23885
negative_passages.text: Structure of a group, $G$, of order $pq$ where $p, q$ are prime.
negative_passages.docid: 115726
negative_passages.text: For $p,q$ prime and $p \neq q$ show that $C_p \times C_q \cong C_{pq}$