query: $a^2=e~\land~a\ast b^2\ast a=b^3\implies b^5=e~~\textrm{for }(G,\ast)~$?
positive_passages.docid: 142263
positive_passages.text: Assuming $ab^2 = b^3a$ and $a^2=1$ prove that the order of $b$ is $5$.
negative_passages.docid: 152741
negative_passages.text: Cardinal of a group $G$ such that for all $x\in G$ we have $x^2=e$
negative_passages.docid: 152604
negative_passages.text: For a finite group of order $2n$ does there exist $x$ such that $x\ast x=e$?
negative_passages.docid: 20571
negative_passages.text: In a monoid, does $x \cdot y=e$ imply $y \cdot x=e$?
negative_passages.docid: 15626
negative_passages.text: Relations of a~b iff b = ak^2
negative_passages.docid: 138911
negative_passages.text: Is $\mathbb R^2$ a field?
negative_passages.docid: 60348
negative_passages.text: How do I prove $F(a)=F(a^2)?$
negative_passages.docid: 13682
negative_passages.text: Is $\langle a,b \mid a^2b^2=1 \rangle$ a semidirect product of $\mathbb{Z}^2$ and $\mathbb{Z}_2$?
negative_passages.docid: 98968
negative_passages.text: $X$ is algebraic over $E$
negative_passages.docid: 52658
negative_passages.text: What are the integer solutions to $a^{b^2} = b^a$ with $a, b \ge 2$
negative_passages.docid: 101506
negative_passages.text: Why is $1, (x-5)^2, (x-5)^3$ a basis of $U=\{p \in \mathcal P_3(\mathbb R) \mid p'(5)=0\}$?
negative_passages.docid: 96458
negative_passages.text: Does $X_n \overset{L^2}{\rightarrow} X$ imply $X_n^2 \overset{L^1}{\rightarrow} X^2$?
negative_passages.docid: 56878
negative_passages.text: Proof that $a\mid b \land b\mid c \Rightarrow a\mid c $
negative_passages.docid: 20679
negative_passages.text: Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.
negative_passages.docid: 76685
negative_passages.text: Determine $\text{Aut}(S)$ where $S = \mathbb{Q}[x]/(x^2)$
negative_passages.docid: 132098
negative_passages.text: Prove $(a^2+b^2)(c^2+d^2)\ge (ac+bd)^2$ for all $a,b,c,d\in\mathbb{R}$.
negative_passages.docid: 73391
negative_passages.text: Prove that if $\gcd(a,b)=1$, then $\gcd(a^2,b^2)=1$