query: In a group G, $a^2b^2=b^2a^2$ and $a^3b^3=b^3a^3$ hold for all $a,b\in G$. Prove that the group is abelian.
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: 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: 135039
negative_passages.text: ${\rm Aut}(G)$ is cyclic $\implies G$ is abelian
negative_passages.docid: 74586
negative_passages.text: Prove that every group of order $4$ is abelian
negative_passages.docid: 158074
negative_passages.text: Cyclic group $G$ with generator $ab$.
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: 26148
negative_passages.text: A group such that $a^m b^m = b^m a^m$ and $a^n b^n = b^n a^n$ ($m$, $n$ coprime) is 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: 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: 48585
negative_passages.text: Proving that if $G/Z(G)$ is cyclic, then $G$ is abelian
negative_passages.docid: 71232
negative_passages.text: if $A$ is Abelian group , $B$ is subgroup of $A$ , Is $B \times A/B \cong A$?
negative_passages.docid: 60208
negative_passages.text: Proof of property of abelian group
negative_passages.docid: 32735
negative_passages.text: If $ \langle G, \star \rangle $ is an abelian group, then for all $a, b \in G$, show that $(a \star b)^{n} = a^{n} \star b^{n}$.
negative_passages.docid: 18182
negative_passages.text: An abelian group $G$ with ${\rm Aut}(G)$ non-abelian
negative_passages.docid: 140334
negative_passages.text: Let $G$ be a group and $H$ be a subgroup of $G$ of index $2$ .Then show that if $a,b\notin H$ then $ab\in H$.
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: 135188
negative_passages.text: Show that if $\vert G\vert = pq$, then either $G$ is abelian or $Z(G) = \{ e\}$.