query: Let $(G, \circ )$ be a group where $a\circ a =e, \ \forall a\in G $. Show that $(G, \circ )$ is abelian.
positive_passages.docid: 146231
positive_passages.text: Let X be a set with an associative binary operation ◦ with identity e. Suppose every element of X satisﬁes x ◦ x = e. Prove that ◦ is commutative.
negative_passages.docid: 97565
negative_passages.text: Quotient group $G/H$ is abelian iff $[G,G] \subseteq H$
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: 76317
negative_passages.text: Let $C$ be the commutator subgroup of $G$. Prove that $G/C$ is abelian
negative_passages.docid: 142375
negative_passages.text: Prove or disprove: If H is a normal subgroup of G such that H and G/H are abelian, then G is abelian.
negative_passages.docid: 116125
negative_passages.text: A group $G$ with $G'$ abelian and every abelian normal subgroup finite
negative_passages.docid: 68971
negative_passages.text: Elementary approach to proving that a group of order 9 is Abelian
negative_passages.docid: 80601
negative_passages.text: Let $a,b$ be in a group $G$. Show $(ab)^n=a^nb^n$ $\forall n\in\mathbb{Z}$ if and only if $ab=ba$.
negative_passages.docid: 94239
negative_passages.text: Proof that if group $G/Z(G)$ is cyclic, then $G$ is commutative
negative_passages.docid: 126893
negative_passages.text: A group of order 2p (p prime) and other conditions - prove abelian.
negative_passages.docid: 63136
negative_passages.text: Prove $G$ is Abelian if $N$ is in the centre of $G$ and $G/N$ is cyclic
negative_passages.docid: 11113
negative_passages.text: Group With an Endomorphism That is "Almost" Abelian is Abelian.
negative_passages.docid: 150585
negative_passages.text: Group with $a=a^{-1}$ for all $a\in G$ is abelian
negative_passages.docid: 25439
negative_passages.text: f(a) = inverse of a is an isomorphism iff a group G is Abelian
negative_passages.docid: 142346
negative_passages.text: The conditions $a^2b^2=b^2a^2$ and $a^3b^3=b^3a^3$ make the group $G$ 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: 20679
negative_passages.text: Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.