query: Prove that the group $G$ is abelian if $a^2 b^2 = b^2 a^2$ and $a^3 b^3 = b^3 a^3$
positive_passages.docid: 142346
positive_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: 117815
negative_passages.text: Non-abelian group $G$ satisfying $(a \cdot b)^i=a^i \cdot b^i,$ for two consecutive integers.
negative_passages.docid: 134403
negative_passages.text: Let $G$ be a group, and let $n\in\mathbb Z$. Show that $(ab)^n=a^nb^n\iff(ab)^{1-n}=a^{1-n}b^{1-n}$.
negative_passages.docid: 158074
negative_passages.text: Cyclic group $G$ with generator $ab$.
negative_passages.docid: 20679
negative_passages.text: Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.
negative_passages.docid: 7015
negative_passages.text: Prove that if $(ab)^i = a^ib^i \forall a,b\in G$ for three consecutive integers $i$ then G is abelian
negative_passages.docid: 108213
negative_passages.text: $N_1,N_2,N_3 \unlhd G, N_i\cap N_j =\{e\}, G = N_iN_j$. Want to show that $G$ is abelian, $N_i$ are isomorphic.
negative_passages.docid: 9110
negative_passages.text: Let G be a group such that $(xy)^2 = (yx)^2$ for all x, y ∈ G. Show that $xy^2 = y^2x$ for all x, y ∈ G.
negative_passages.docid: 37808
negative_passages.text: Show that $\gcd(a^2, b^2) = \gcd(a,b)^2$
negative_passages.docid: 109611
negative_passages.text: If G is a group, prove that $(ab)^n =a^nb^n$ implies G is abelian.
negative_passages.docid: 87882
negative_passages.text: If $N \lhd G$ is abelian and $G/N$ is abelian then $G$ is abelian
negative_passages.docid: 138894
negative_passages.text: Prove that if $3\mid(a^2+b^2)$,then $3\mid a$ and $ 3\mid b$
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: 59128
negative_passages.text: If $|\lbrace g \in G: \pi (g)=g^{-1} \rbrace|>\frac{3|G|}{4}$, then $G$ is an abelian group.
negative_passages.docid: 97565
negative_passages.text: Quotient group $G/H$ is abelian iff $[G,G] \subseteq H$
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: 31380
negative_passages.text: Exponent and abelian groups