query: Let G be a group and p and q be primes.
positive_passages.docid: 141890
positive_passages.text: $G$ is a Group of order $pq$, if $G$ has exactly one subgroup of order $p$ and another with order $q$, then $G$ is cyclic ($p,q$ are prime)
negative_passages.docid: 47239
negative_passages.text: For $p,q$ primes such that $q|p-1$, show there is only one non-abelian group of order $pq$, up to isomorphism
negative_passages.docid: 34567
negative_passages.text: If $G_1$ and $G_2$ are finite group and $H\leq G_1\times G_2$ then $H=P_1\times P_2$ with $P_i\leq G_i$?
negative_passages.docid: 126893
negative_passages.text: A group of order 2p (p prime) and other conditions - prove abelian.
negative_passages.docid: 99205
negative_passages.text: Group of order $p^k q$ where $p$ and $q$ are distinct primes - how to prove this property?
negative_passages.docid: 64269
negative_passages.text: Finding the minimal $n$ such that a given finite group $G$ is a subgroup of $S_n$
negative_passages.docid: 104080
negative_passages.text: Group with order $pq$ has subgroups of order $p$ and $q$
negative_passages.docid: 77221
negative_passages.text: Prove that $G$ has a subgroup isomorphic to $G/H$.
negative_passages.docid: 71420
negative_passages.text: Prove that a group is abelian.
negative_passages.docid: 118337
negative_passages.text: Let $S$ be a normal p-subgroup of a finite group $G$. Prove that $S \subseteq P$ for every Sylow p-subgroup $P$ of $G$
negative_passages.docid: 57427
negative_passages.text: Group of order $8p$ is solvable, for any prime $p$
negative_passages.docid: 56650
negative_passages.text: Prove that $(G, \circ)$ is a group if $a\circ x = b$ and $x\circ a = b$ have unique solutions
negative_passages.docid: 102979
negative_passages.text: If $[G:H]$ and $[G:K]$ are relatively prime, then $G=HK$
negative_passages.docid: 22742
negative_passages.text: Prove that if $p$ and $p^2+2$ are prime then $p^3+2$ is prime too
negative_passages.docid: 95534
negative_passages.text: If $|G|=p^nq$, then $G$ contains a unique normal subgroup of index $q$
negative_passages.docid: 52121
negative_passages.text: Let $G$ be a finite group with $|G|>2$. Prove that ${\rm Aut}(G)$ contains at least two elements.
negative_passages.docid: 135701
negative_passages.text: An example of a group such that $G \cong G \times G$