query: $a^2b^2=b^2a^2 , a^3b^3=b^3a^3$ , then $G$ 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: 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: 141125
negative_passages.text: Group with order $p^2$ must be abelian . How to prove that?
negative_passages.docid: 3756
negative_passages.text: If $G$ is abelian, then the set of all $g \in G$ such that $g = g^{-1}$ is a subgroup of $G$
negative_passages.docid: 3838
negative_passages.text: If a group satisfies $x^3=1$ for all $x$, is it necessarily abelian?
negative_passages.docid: 105180
negative_passages.text: When does $(ab)^n = a^n b^n$ imply a group is abelian?
negative_passages.docid: 82938
negative_passages.text: $\gcd(a,b)=1, x^a = y^b\Rightarrow x = n^b$, $ y = n^a$ for an integer $n$.
negative_passages.docid: 66313
negative_passages.text: $\mathbb{Z}[i]/(a+bi) \cong \mathbb{Z}_{a^2+b^2}$ if $(a,b)=1$
negative_passages.docid: 80601
negative_passages.text: Let $a,b$ be in a group $G$. Show $(ab)^n=a^nb^n$ $\forall n\in\mathbb{Z}$ if and only if $ab=ba$.
negative_passages.docid: 76507
negative_passages.text: $\gcd(b^x - 1, b^y - 1, b^ z- 1,\dots) = b^{\gcd(x, y, z,\dots)} -1$
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: 2208
negative_passages.text: $\gcd(a,b,c)=\gcd(\gcd(a,b),c)\,$ [GCD Associative Law]
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: 34330
negative_passages.text: How to prove $\gcd(a^2,b^2) = (\gcd(a,b))^2$?
negative_passages.docid: 17032
negative_passages.text: If $ab=ba$, Prove $a^2$ commutes with $b^2$
negative_passages.docid: 52357
negative_passages.text: If $a,b \in G$, show that $ab^{n}a^{-1}=(aba^{-1})^{n}$.
negative_passages.docid: 109611
negative_passages.text: If G is a group, prove that $(ab)^n =a^nb^n$ implies G is abelian.