query: Existence of elements of order $2$ in $S_n$ whose Product has order $n$
positive_passages.docid: 28266
positive_passages.text: For $n\geq 3$ there exist $x,y\in S_n$ such that $x$ and $y$ have order $2$ and $xy$ has order $n$.
negative_passages.docid: 46687
negative_passages.text: Elements of $GL_{2}(\mathbb{Z})$ of finite order
negative_passages.docid: 57221
negative_passages.text: counting the number of elements in a conjugacy class of $S_n$
negative_passages.docid: 91774
negative_passages.text: Existence of group $G$ with $|G|>2$ and automorphism group of odd order
negative_passages.docid: 107010
negative_passages.text: How many elements of order $k$ are in $S_n$?
negative_passages.docid: 43058
negative_passages.text: If $n > 2$, prove that the order of the multiplicative group of units modulo n, $U_n$, is even.
negative_passages.docid: 9651
negative_passages.text: Cardinality of cartesian product of an infinite set with N
negative_passages.docid: 85579
negative_passages.text: $G$ a group s.t. every non-identity element has order 2. If $G$ is finite, prove $|G| = 2^n$ and $G \simeq C_2 \times C_2 \times\cdots\times C_2$
negative_passages.docid: 100061
negative_passages.text: The Maximum possible order for an element $S_n$
negative_passages.docid: 143550
negative_passages.text: Order of product of two elements in an abelian group
negative_passages.docid: 46260
negative_passages.text: Prove that a group generated by two elements of order $2$, $x$ and $y$, is isomorphic to $D_{2n}$, where $n = |xy|.$
negative_passages.docid: 104513
negative_passages.text: Order of elements in $Z_n$
negative_passages.docid: 5089
negative_passages.text: An element of a group has the same order as its inverse
negative_passages.docid: 48393
negative_passages.text: Abelian groups of order n.
negative_passages.docid: 30676
negative_passages.text: $G'$ and the product of all the elements in an odd order group
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: 119808
negative_passages.text: Proving that $n$-cycle has order $n$