query: Suppose $x^2=1$ for all $x \in G$. Prove that $G$ is abelian.
positive_passages.docid: 150585
positive_passages.text: Group with $a=a^{-1}$ for all $a\in G$ is abelian
negative_passages.docid: 55976
negative_passages.text: Are groups satisfying $g=g^{-1}$ for all $g \in G$ abelian?
negative_passages.docid: 69279
negative_passages.text: If $\gcd(a,b)=1$ then, $\gcd(a^2,b^2)=1$
negative_passages.docid: 106577
negative_passages.text: If G is not abelian with order $p^3$ then $G'=Z(G)$
negative_passages.docid: 100605
negative_passages.text: let $G$ to be finite abelian group of order O(G), let n to be prime number and (O(G),n)=1 prove that $g=x^n$ for any $ x \in G$
negative_passages.docid: 106809
negative_passages.text: $f: G→G$ defined by $f(x) =x^2$ is a homomorphism if and only if $G$ is abelian.
negative_passages.docid: 86823
negative_passages.text: proof verification: If $f:G \rightarrow H$ is group homomorphism, and $H$ is abelian, then $G$ is abelian
negative_passages.docid: 26624
negative_passages.text: Prove that is group is abelian
negative_passages.docid: 103801
negative_passages.text: $xT=x$ iff $x=e$ implies that every $g$ may be written as $x^{-1}(xT)$ for some $x\in G$.
negative_passages.docid: 147398
negative_passages.text: If every element of $H$ and $G/H$ is a square, then prove that so is every element of $G$.
negative_passages.docid: 127603
negative_passages.text: If $G$ be a group of order $8$ and $x$ be a element of order $4$. Prove that $x^2 \in Z(G)$
negative_passages.docid: 4838
negative_passages.text: For $N\unlhd G$ , with $C_G(N)\subset N$ we have $G/N$ is abelian
negative_passages.docid: 125640
negative_passages.text: Group of order $p^2$ is abelian.
negative_passages.docid: 138406
negative_passages.text: If N and every subgroup of N is normal in G then G/N is abelian .
negative_passages.docid: 68971
negative_passages.text: Elementary approach to proving that a group of order 9 is Abelian
negative_passages.docid: 126893
negative_passages.text: A group of order 2p (p prime) and other conditions - prove abelian.
negative_passages.docid: 82037
negative_passages.text: Suppose $G$ is a group generated by elements $x$ and $y$ where $xy^2 = y^3x$ and $yx^3 = x^2y$ What can you prove about $G$?