query: Group of even order contains an element of order 2
positive_passages.docid: 152604
positive_passages.text: For a finite group of order $2n$ does there exist $x$ such that $x\ast x=e$?
negative_passages.docid: 69833
negative_passages.text: Number of elements not equal to their inverses is even number
negative_passages.docid: 92750
negative_passages.text: Groups where all elements are order 3
negative_passages.docid: 102634
negative_passages.text: Order of the Rubik's cube group
negative_passages.docid: 57427
negative_passages.text: Group of order $8p$ is solvable, for any prime $p$
negative_passages.docid: 134676
negative_passages.text: Number of elements of order $p$ is a multiple of $p-1$ (finite group).
negative_passages.docid: 143550
negative_passages.text: Order of product of two elements in an abelian group
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: 79055
negative_passages.text: Orders of a symmetric group
negative_passages.docid: 13153
negative_passages.text: at least one element fixed by all the group
negative_passages.docid: 2794
negative_passages.text: Prove that if a group contains exactly one element of order 2, then that element is in the center of the group.
negative_passages.docid: 8742
negative_passages.text: Examples and further results about the order of the product of two elements in a group
negative_passages.docid: 67554
negative_passages.text: In a group, the number of non-self-inverse elements is even
negative_passages.docid: 51196
negative_passages.text: A group of order $p^2q^2$ is never simple
negative_passages.docid: 128834
negative_passages.text: Proving the number of even and odd permutations of a subgroup $H<S_{n}$ are equal, provided $H$ is not contained in $A_{n}$
negative_passages.docid: 89508
negative_passages.text: Existence of normal subgroups for a group of order $36$
negative_passages.docid: 38083
negative_passages.text: Order of conjugate of an element given the order of its conjugate