query: Let $a,b\in G$ elements of order $5$. If $a^3=b^3$ then $a=b$.
positive_passages.docid: 150958
positive_passages.text: Group theory question. If $m$ and $n$ are relatively prime, and $a$, $b$ belong to group and $a^m = b^m$ and $a^n = b^n$, how would one prove $a = b$?
negative_passages.docid: 15191
negative_passages.text: If $a^2$ divides $b^2$, then $a$ divides $b$
negative_passages.docid: 71012
negative_passages.text: If $a, b$ are positive integers such that $a|b^2, b^2|a^3,....$ then $a=b$.
negative_passages.docid: 90916
negative_passages.text: Prove that if $3\mid a^2+b^2$ then $3\mid a$ and $3\mid b$.
negative_passages.docid: 117468
negative_passages.text: $o(g^{k})=\frac{o(g)}{\gcd(o(g),k)}$ order of a power of a group element.
negative_passages.docid: 99630
negative_passages.text: For every $g\in G$ there exists an $h\in G$ such that $g = h^3$
negative_passages.docid: 21006
negative_passages.text: Let $a,b$ be positive integers such that $a\mid b^2 , b^2\mid a^3 , a^3\mid b^4 \ldots$ so on , then $a=b$?
negative_passages.docid: 2289
negative_passages.text: Prove that if $\gcd( a, b ) = 1$ then $\gcd( ac, b ) = \gcd( c, b ) $
negative_passages.docid: 49299
negative_passages.text: Proof by Induction: For any integer $n > 0$, $3|(5^{2n} − 1)$.
negative_passages.docid: 156319
negative_passages.text: Show that a group with $|G| = 33$ contains an element of order $3$
negative_passages.docid: 121142
negative_passages.text: Take $G$ to be a group of order $600$. Prove that for any element a in $G$ there exist a $b$ in $G$ such that $a = b^7$
negative_passages.docid: 43059
negative_passages.text: Show that $g^k=e$ if and only if the order of $g$ is finite and divides $k$.
negative_passages.docid: 94516
negative_passages.text: If $3$ divides $a^2 + b^2$, then $3$ divides $a$ and $3$ divides $b$
negative_passages.docid: 134403
negative_passages.text: Let $G$ be a group, and let $n\in\mathbb Z$. Show that $(ab)^n=a^nb^n\iff(ab)^{1-n}=a^{1-n}b^{1-n}$.
negative_passages.docid: 45760
negative_passages.text: if G is generated by {a,b} and ab=ba, then prove G is Abelian
negative_passages.docid: 7416
negative_passages.text: Let $ A $, $ B $ and $ C $ be sets. If $ A \cup B = A \cup C $ and $ A \cap B = A \cap C $, then show that $ B = C $.
negative_passages.docid: 126358
negative_passages.text: Prove for $\gcd$ of two integers $a,b$, that $(a^n,b^n) = (a,b)^n$