query: If $g^2 = e$ for all $g \in G$, then $G$ is abelian
positive_passages.docid: 20679
positive_passages.text: Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.
negative_passages.docid: 135188
negative_passages.text: Show that if $\vert G\vert = pq$, then either $G$ is abelian or $Z(G) = \{ e\}$.
negative_passages.docid: 116125
negative_passages.text: A group $G$ with $G'$ abelian and every abelian normal subgroup finite
negative_passages.docid: 133627
negative_passages.text: If $\phi(g)=g^3$ is a homomorphism and $3 \nmid |G|$, $G$ is abelian.
negative_passages.docid: 75127
negative_passages.text: Group $G$ of order $p^2$: $\;G\cong \mathbb Z_{p^2}$ or $G\cong \mathbb Z_p \times \mathbb Z_p$
negative_passages.docid: 45760
negative_passages.text: if G is generated by {a,b} and ab=ba, then prove G is Abelian
negative_passages.docid: 57809
negative_passages.text: For which $n$, $G$ is abelian?
negative_passages.docid: 6153
negative_passages.text: Suppose that $G$ is a group with the property that for every choice of elements in $G$, $axb=cxd$ implies $ab=cd$. Prove that $G$ is Abelian.
negative_passages.docid: 66510
negative_passages.text: A group is abelian if and only if the center of the group is all the group
negative_passages.docid: 15633
negative_passages.text: If G is a group of even order, prove it has an element $a \neq e$ satisfying $a^2 = e$.
negative_passages.docid: 65900
negative_passages.text: Show that a finite group with certain automorphism is abelian
negative_passages.docid: 125640
negative_passages.text: Group of order $p^2$ is abelian.
negative_passages.docid: 68733
negative_passages.text: If $G/Z(G)$ is cyclic then $G$ is abelian – what is the point?
negative_passages.docid: 48585
negative_passages.text: Proving that if $G/Z(G)$ is cyclic, then $G$ is abelian
negative_passages.docid: 17510
negative_passages.text: Proving a set is an abelian group.
negative_passages.docid: 152071
negative_passages.text: Group $G$ is cyclic $\iff$ every subgroup of $G$ has the form $G^k$, where $G^k=\{g^k\mid g\in G\}$
negative_passages.docid: 94507
negative_passages.text: Given $G$ group and $H \le G$ such that $|G:H|=2$, how does $x^2 \in H$ for every $x \in G$?