query: Let $A$ be a group, where $a^2=1$, a belongs to $A$. Prove that this group is commutative.
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: 87946
negative_passages.text: Proving that the unities of a ring form a group under multiplication
negative_passages.docid: 18971
negative_passages.text: Show that a ring is commutative if it has the property that ab = ca implies b = c when $a\neq 0$
negative_passages.docid: 111886
negative_passages.text: Let $(G,•)$ be a group in which $(a•b)^3 = a^3•b^3$. Prove that $H=\{x^3 : x \in G\}$ is a normal subgroup of $G$.
negative_passages.docid: 79933
negative_passages.text: Let $G$ be a group and $c\in G$. Define $*$ on $G$ by: $x*y=xdy$ for all $x,y\in G$, where $d$ is the inverse of $c$. Prove $(G,*)$ is a group.
negative_passages.docid: 116707
negative_passages.text: Prove or disprove: $(\mathbb{Z}^*, \cdot)$ and/or $(\mathbb{Z}^*, \div)$ is a group.
negative_passages.docid: 150585
negative_passages.text: Group with $a=a^{-1}$ for all $a\in G$ is abelian
negative_passages.docid: 13631
negative_passages.text: Showing group with $p^2$ elements is Abelian
negative_passages.docid: 103216
negative_passages.text: The set $\{g^2 | g \in G\}$ in a group $G$
negative_passages.docid: 59561
negative_passages.text: Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple
negative_passages.docid: 156429
negative_passages.text: If $G$ is a group whereby $(a\cdot b)^{i} =a^i\cdot b^i$ for three consecutive integers $i$ for all $a, b \in G$, show $G$ is abelian.
negative_passages.docid: 150042
negative_passages.text: Prove $x^{-1}Hx$ is a subgroup of $G$
negative_passages.docid: 144988
negative_passages.text: Commutator Subgroup of Direct Product equals the Direct Product of the Commutator Subgroups
negative_passages.docid: 66313
negative_passages.text: $\mathbb{Z}[i]/(a+bi) \cong \mathbb{Z}_{a^2+b^2}$ if $(a,b)=1$
negative_passages.docid: 90811
negative_passages.text: If $[G' : G'']\leq p^2$, then $G'$ is abelian.
negative_passages.docid: 79452
negative_passages.text: How to prove that it is a group?
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 $ab=ba$.