query: Let $G=G_1\times G_2$, prove $G'=G_1'\times G_2'$
positive_passages.docid: 144988
positive_passages.text: Commutator Subgroup of Direct Product equals the Direct Product of the Commutator Subgroups
negative_passages.docid: 82037
negative_passages.text: Suppose $G$ is a group generated by elements $x$ and $y$ where $xy^2 = y^3x$ and $yx^3 = x^2y$ What can you prove about $G$?
negative_passages.docid: 80273
negative_passages.text: Prove the centralizer of an element in group $G$ is a subgroup of $G$
negative_passages.docid: 13253
negative_passages.text: Prove $f(S \cup T) = f(S) \cup f(T)$
negative_passages.docid: 73391
negative_passages.text: Prove that if $\gcd(a,b)=1$, then $\gcd(a^2,b^2)=1$
negative_passages.docid: 150042
negative_passages.text: Prove $x^{-1}Hx$ is a subgroup of $G$
negative_passages.docid: 101872
negative_passages.text: Prove that $\gcd(a^2, b^2) = \gcd(a, b)^2$
negative_passages.docid: 75604
negative_passages.text: Proof that $\partial(A\times B) = ((\partial A)\times B)\cup (A\times(\partial B))$
negative_passages.docid: 110313
negative_passages.text: Prove that $\gcd(f(x),g(x)) = 1$.
negative_passages.docid: 107716
negative_passages.text: Show that $G$ is cyclic
negative_passages.docid: 110103
negative_passages.text: Prove that there are $0<c_1 < c_2<1$ s.t. $\frac{1}{g'(c_1)}+\frac{1}{g'(c_2)}=2$ . The same for $g'(c_1)g'(c_2)=1$.
negative_passages.docid: 149553
negative_passages.text: Is it always true that $(A_1 \cup A_2) \times (B_1 \cup B_2)=(A_1\times B_1) \cup (A_2 \times B_2)$
negative_passages.docid: 139111
negative_passages.text: Prove that $a<c<b$, $\frac{f(b)-f(a)-(b-a)f'(a)}{g(b)-g(a)-(b-a)g'(a)}=\frac{f"(c)}{g"(c)}$
negative_passages.docid: 96329
negative_passages.text: Show that for any $k$-cycle $(a_1 a_2...a_k) \in S_n$, and for any permutation $\pi \in S_n$, $\pi(a_1...a_k)\pi^{-1}=(\pi(a_1)..\pi(a_k))$
negative_passages.docid: 10024
negative_passages.text: How to prove $(f \circ\ g) ^{-1} = g^{-1} \circ\ f^{-1}$? (inverse of composition)
negative_passages.docid: 71245
negative_passages.text: $2 \times 2 $ Orthogonal Group
negative_passages.docid: 20679
negative_passages.text: Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.