query: G is solvable iff factors have prime order
positive_passages.docid: 22248
positive_passages.text: $G$ solvable $\implies$ composition factors of $G$ are of prime order.
negative_passages.docid: 70001
negative_passages.text: Any group of order $2p$ has a subgroup of order $p~$($p$ being a prime)
negative_passages.docid: 36147
negative_passages.text: A finite group is nilpotent iff two elements with relatively prime order commute
negative_passages.docid: 4402
negative_passages.text: Group of order $pqr$, $p < q < r$ primes
negative_passages.docid: 116080
negative_passages.text: Let G be a finite group with more than one element. Show that G has an element of prime order
negative_passages.docid: 43591
negative_passages.text: How to factor numbers that are the product of two primes
negative_passages.docid: 39843
negative_passages.text: Let $P$ be a prime. Show that $\exists$ $x \in \mathbb{N}$ such that $f(x) = p$ then $\exists$ $y \in \mathbb{N}$ such that $g(y) = p$
negative_passages.docid: 3703
negative_passages.text: Order of finite fields is $p^n$
negative_passages.docid: 126098
negative_passages.text: Isomorphism of factor groups
negative_passages.docid: 148904
negative_passages.text: Groups of order $p^3$
negative_passages.docid: 139218
negative_passages.text: Is this group solvable?
negative_passages.docid: 116361
negative_passages.text: Relationship between prime factorizations of $n$ and $n+1$?
negative_passages.docid: 19668
negative_passages.text: Does every group whose order is a power of a prime p contain an element of order p?
negative_passages.docid: 125330
negative_passages.text: How to find element g order q, given 2 large primes p and q where q|p-1
negative_passages.docid: 28362
negative_passages.text: Product of two cyclic groups is cyclic iff their orders are co-prime
negative_passages.docid: 108830
negative_passages.text: Let $G$ be a group of order $2p$ , where $p$ is a prime greater than $2$. Then, G is isomorphic to $\mathbb{Z}_{2p}$ or $D_p$
negative_passages.docid: 121481
negative_passages.text: Non-abelian finite $p$-group in which every element has prime order?