query: Prove that $\{(1 2),(1 2 ... n)\}$ can generate $S_n$
positive_passages.docid: 14139
positive_passages.text: $(12)$ and $(123\dots n)$ are generators of $S_n$
negative_passages.docid: 138819
negative_passages.text: Prove that $\phi(n) \geq \sqrt{n}/2$
negative_passages.docid: 149609
negative_passages.text: Conditions on cycle types for permutations to generate $S_n$
negative_passages.docid: 12020
negative_passages.text: Prove $\sqrt{n+1}-\sqrt{n}\lt \frac{1}{2\sqrt{n}}$
negative_passages.docid: 103106
negative_passages.text: Proving that any permutation in $S_n$ can be written as a product of disjoint cycles
negative_passages.docid: 113411
negative_passages.text: If $E$ is generated by $E_1$ and $E_2$, then $[E:F]\leq [E_1:F][E_2:F]$?
negative_passages.docid: 69921
negative_passages.text: Prove that: $(1+a_1)(1+a_2)...(1+a_n) \le 1 + S_n + (S_n)^2/2! + ... + (S_n)^n/n!$
negative_passages.docid: 40692
negative_passages.text: Proving by strong induction that $\forall n \ge 2, \;\forall d \ge 2 : d \mid n(n+1)(n+2)...(n+d-1) $
negative_passages.docid: 121544
negative_passages.text: Proof by Induction: $2(\sqrt{n+1} - \sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n}-\sqrt{n-1})$
negative_passages.docid: 151520
negative_passages.text: About the parity of the product $(a_1-1)(a_2-2)\cdots(a_n-n)$
negative_passages.docid: 18141
negative_passages.text: Embedding $S_n$ into $A_{n+2}$
negative_passages.docid: 29065
negative_passages.text: Encode each $n_1,n_2,n_3,...∈N^N$ by an inﬁnite sequence of 0s and 1s with inﬁnitely many 0s, and give a proof that $N^N$ is equinumerous with $R$.
negative_passages.docid: 128566
negative_passages.text: Proving $2$ generates $(\mathbb{Z}/19\mathbb{Z})^*$ by only looking at a few powers
negative_passages.docid: 71014
negative_passages.text: Prove that 2(nC0+nC2...nC$\frac{(N-1)}{2}$=$2^n$
negative_passages.docid: 148285
negative_passages.text: Generator of group $D_n$
negative_passages.docid: 88425
negative_passages.text: Prove that $\,\sqrt [n] n < 1 + \sqrt{\frac{2}{n}}\,$
negative_passages.docid: 127720
negative_passages.text: Prove that $\bigcup_{n \in \Bbb{N}} [1/n,1] = (0,1]$.