query: If $a$ is the only element of order $2$ in a group, show that $a$ is in the center of the group.
positive_passages.docid: 2794
positive_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: 63359
negative_passages.text: Show that the product of all group elements is $e$ or $u$ if $G$ has exactly one involution $u$
negative_passages.docid: 41977
negative_passages.text: If $G$ is a group, show that $x^2ax=a^{-1}$ has a solution if and only if $a$ is a cube in $G$
negative_passages.docid: 7121
negative_passages.text: If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.
negative_passages.docid: 15526
negative_passages.text: Question on groups of order $pq$
negative_passages.docid: 1364
negative_passages.text: Let $G$ be a group in which $a^2=e$ for all elements of $a$ of $G$. Show that $G$ is Abelian.
negative_passages.docid: 40846
negative_passages.text: When can a (finite) group be written as the quotient of some other group by its center?
negative_passages.docid: 101543
negative_passages.text: Let $A$ be a group, where $a^2=1$, a belongs to $A$. Prove that this group is commutative.
negative_passages.docid: 60982
negative_passages.text: Group where every element is order 2
negative_passages.docid: 117468
negative_passages.text: $o(g^{k})=\frac{o(g)}{\gcd(o(g),k)}$ order of a power of a group element.
negative_passages.docid: 96123
negative_passages.text: Element of maximal order in alternating group $A_{10}$: how to prove this is it?
negative_passages.docid: 80273
negative_passages.text: Prove the centralizer of an element in group $G$ is a subgroup of $G$
negative_passages.docid: 24181
negative_passages.text: Showing that $G$ is a group under an alternative operation.
negative_passages.docid: 1577
negative_passages.text: Order of nontrivial elements is 2 implies Abelian group
negative_passages.docid: 51373
negative_passages.text: $G$ non abelian, order $p^3$ ($p$ prime). Suppose that the center is $p^2$, prove that $\exists\ x$ outside of the center, of order p
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: 30676
negative_passages.text: $G'$ and the product of all the elements in an odd order group