query: Show that a group $G$ is abelian if $(gh)^3 = g^3h^3$, $(gh)^4 = g^4h^4$, and $(gh)^5 = g^5h^5$ for all $g$ and $h$ in $G$.
positive_passages.docid: 156429
positive_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: 106577
negative_passages.text: If G is not abelian with order $p^3$ then $G'=Z(G)$
negative_passages.docid: 3907
negative_passages.text: Abelian quotient group
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: 46342
negative_passages.text: Prove that $G$ is abelian iff $\varphi(g) = g^2$ is a homomorphism
negative_passages.docid: 82037
negative_passages.text: Suppose $G$ is a group generated by elements $x$ and $y$ where $xy^2 = y^3x$ and $yx^3 = x^2y$ What can you prove about $G$?
negative_passages.docid: 124709
negative_passages.text: Suppose G be a non-abelian group and H,K be two abelian subgroups of G. Then must HK be an abelian subgroup of G?
negative_passages.docid: 11003
negative_passages.text: Proving that $G/N$ is an abelian group
negative_passages.docid: 26624
negative_passages.text: Prove that is group is abelian
negative_passages.docid: 17510
negative_passages.text: Proving a set is an abelian group.
negative_passages.docid: 146760
negative_passages.text: Prove that H is a subgroup of G
negative_passages.docid: 90811
negative_passages.text: If $[G' : G'']\leq p^2$, then $G'$ is abelian.
negative_passages.docid: 11825
negative_passages.text: Let $H$ be a subgroup of $G$. Let $K = \{x \in G: xax^{-1} \in H \iff a \in H\}$. Prove that $K$ is a subgroup of $G$.
negative_passages.docid: 36887
negative_passages.text: Suppose $G$ is a group of order 4. Show either $G$ is cyclic or $x^2=e$.
negative_passages.docid: 37414
negative_passages.text: Question on Groups $G=\langle x,y|x^4=y^4=e,xyxy^{-1}=e\rangle$
negative_passages.docid: 86823
negative_passages.text: proof verification: If $f:G \rightarrow H$ is group homomorphism, and $H$ is abelian, then $G$ is abelian
negative_passages.docid: 138406
negative_passages.text: If N and every subgroup of N is normal in G then G/N is abelian .