query: Let $(G,*)$ be a group with $(a*b)^3=a^3*b^3$ and $(a*b)^5=a^5*b^5\forall a,b∈G$. Show that $(G,*)$ is abelian.
positive_passages.docid: 27905
positive_passages.text: Let $G$ be a group, where $(ab)^3=a^3b^3$ and $(ab)^5=a^5b^5$. Prove that $G $ is an abelian group.
negative_passages.docid: 108425
negative_passages.text: Problem on abelian group
negative_passages.docid: 33233
negative_passages.text: Need to prove that $(S,\cdot)$ defined by the binary operation $a\cdot b = a+b+ab$ is an abelian group on $S = \Bbb R \setminus \{-1\}$.
negative_passages.docid: 133627
negative_passages.text: If $\phi(g)=g^3$ is a homomorphism and $3 \nmid |G|$, $G$ is abelian.
negative_passages.docid: 43580
negative_passages.text: Let $p$ be a prime number and $G$ a non abelian group or order $p^3$. Prove that $Z(G) = [G,G]$.
negative_passages.docid: 49527
negative_passages.text: Prove that $a$ commutes with each of its conjugates in $G$ iff $a$ belongs to an abelian normal subgroup of $G$.
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: 52357
negative_passages.text: If $a,b \in G$, show that $ab^{n}a^{-1}=(aba^{-1})^{n}$.
negative_passages.docid: 57421
negative_passages.text: If $\forall x \in G$ where G is a group $x=x^{-1}$ then how is this group abelian?
negative_passages.docid: 13834
negative_passages.text: Show that G is a group, if G is finite, the operation is associative, and cancellation law holds
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: 106577
negative_passages.text: If G is not abelian with order $p^3$ then $G'=Z(G)$
negative_passages.docid: 31380
negative_passages.text: Exponent and abelian groups
negative_passages.docid: 143550
negative_passages.text: Order of product of two elements in an abelian group
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: 102742
negative_passages.text: order $a$ = 5, $a^3b = ba^3$. show that that $ab = ba$.
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.