query: If $a^pb^p=b^pa^p$ and $a^qb^q=b^qa^q$ for coprime $p,q$ and all $a,b\in G$, then $G$ is Abelian
positive_passages.docid: 26148
positive_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: 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: 1364
negative_passages.text: Let $G$ be a group in which $a^2=e$ for all elements of $a$ of $G$. Show that $G$ is Abelian.
negative_passages.docid: 135188
negative_passages.text: Show that if $\vert G\vert = pq$, then either $G$ is abelian or $Z(G) = \{ e\}$.
negative_passages.docid: 68732
negative_passages.text: Prove that a semigroup satisfying $a^pb^q=ba$ is commutative
negative_passages.docid: 13631
negative_passages.text: Showing group with $p^2$ elements is Abelian
negative_passages.docid: 108425
negative_passages.text: Problem on abelian group
negative_passages.docid: 90811
negative_passages.text: If $[G' : G'']\leq p^2$, then $G'$ is abelian.
negative_passages.docid: 141125
negative_passages.text: Group with order $p^2$ must be abelian . How to prove that?
negative_passages.docid: 7091
negative_passages.text: If $\gcd(a,b)=1$, then $\gcd(a^n,b^n)=1$
negative_passages.docid: 108310
negative_passages.text: $(a_1 a_2 \cdots a_n)^2=e$ in a finite abelian group
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: 145159
negative_passages.text: Finite group is abelian if the representatives of its conjugacy classes commute
negative_passages.docid: 48585
negative_passages.text: Proving that if $G/Z(G)$ is cyclic, then $G$ is abelian
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: 40287
negative_passages.text: Prove: if $a,b\in G$ commute with probability $>5/8$, then $G$ is abelian
negative_passages.docid: 55976
negative_passages.text: Are groups satisfying $g=g^{-1}$ for all $g \in G$ abelian?