query: Show that if $n\geq5$, then every element of $S_n$ is a product of 4-cycles.
positive_passages.docid: 19026
positive_passages.text: What do all the $k$-cycles in $S_n$ generate?
negative_passages.docid: 91772
negative_passages.text: Proving that $A_n$ is the only proper nontrivial normal subgroup of $S_n$, $n\geq 5$
negative_passages.docid: 48081
negative_passages.text: Prove that for all $n \in \mathbb{N}$: $2^{n-1}(a^n + b^n) \ge (a + b)^n$
negative_passages.docid: 147253
negative_passages.text: Prove for any $K \leq G$, either $K \leq N$ or $KN=G$ and $[K:K \cap N]=p$ where N is normal in G and $[G:N]=p$
negative_passages.docid: 149609
negative_passages.text: Conditions on cycle types for permutations to generate $S_n$
negative_passages.docid: 35800
negative_passages.text: If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
negative_passages.docid: 107010
negative_passages.text: How many elements of order $k$ are in $S_n$?
negative_passages.docid: 70066
negative_passages.text: Show that product of primes, $\prod_{k=1}^{\pi(n)} p_k < 4^n$
negative_passages.docid: 68201
negative_passages.text: Cyclic group of order $n$
negative_passages.docid: 23544
negative_passages.text: Complete induction proof that every $n > 1$ can be written as a product of primes
negative_passages.docid: 100987
negative_passages.text: Let f be an r-cycle in $S_n$. Given any $h \in S_n$, show that $hfh^{-1} = (h(x_1), h(x_2), \dots, h(x_r))$
negative_passages.docid: 54554
negative_passages.text: If a product of relatively prime integers is an $n$th power, then each is an $n$th power
negative_passages.docid: 146969
negative_passages.text: Proving that every group of order $4$ is isomorphic to $\Bbb Z_4$ or $\Bbb Z^*_8$
negative_passages.docid: 50762
negative_passages.text: If $S_n$ has cyclic subgroups of these orders or not?
negative_passages.docid: 47179
negative_passages.text: Prove that $[N, G']=1$ when $N$ is a nilpotent subgroup of a group $G$?
negative_passages.docid: 1507
negative_passages.text: The product of $n$ consecutive integers is divisible by $n$ factorial
negative_passages.docid: 44480
negative_passages.text: Let $\sigma :\{1,2,3,4,5\} \rightarrow \{1,2,3,4,5\}$ be permutations such that $ \sigma^{-1}(j) \leq \sigma(j)~\forall j, 1 \leq j \leq 5$.