query: Prove/Disprove: If $K\leq G \times H$ Then $K\in G'\times H'$ Where $G'\leq G$ And $H'\leq H$
positive_passages.docid: 151862
positive_passages.text: Is any subgroup of a direct product isomorphic to a direct product of subgroups?
negative_passages.docid: 94507
negative_passages.text: Given $G$ group and $H \le G$ such that $|G:H|=2$, how does $x^2 \in H$ for every $x \in G$?
negative_passages.docid: 13330
negative_passages.text: Inductive Proof that $k!<k^k$, for $k\geq 2$.
negative_passages.docid: 63267
negative_passages.text: Under what conditions on the sets $A$ and $B$ will it be true that $A\times B= B\times A$
negative_passages.docid: 151621
negative_passages.text: $G$ a group and $H, K\mathrel{\unlhd}G$. Assuming that $H \cap K = \{1_G\}$ and $G = \langle H, K \rangle$, prove that $G \cong H \times K$
negative_passages.docid: 141683
negative_passages.text: $ |G_1 |$ and $|G_2 | $ are coprime. Show that $K = H_1 \times H_2$
negative_passages.docid: 64096
negative_passages.text: Prove that $||z_1|-|z_2|| \leq |z_1-z_2|$.
negative_passages.docid: 147253
negative_passages.text: Prove for any $K \leq G$, either $K \leq N$ or $KN=G$ and $[K:K \cap N]=p$ where N is normal in G and $[G:N]=p$
negative_passages.docid: 29996
negative_passages.text: Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
negative_passages.docid: 95686
negative_passages.text: $Lie(G \times H)\cong Lie(G)\oplus Lie(H)$
negative_passages.docid: 106779
negative_passages.text: How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
negative_passages.docid: 148668
negative_passages.text: How to show $A\times (B\cup C) = (A\times B)\cup (A\times C)$
negative_passages.docid: 59566
negative_passages.text: Proving that $ka \equiv kb \pmod{m} \iff a \equiv b \pmod{m/\gcd(k,m)}$
negative_passages.docid: 135808
negative_passages.text: Show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$
negative_passages.docid: 46355
negative_passages.text: If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?
negative_passages.docid: 86336
negative_passages.text: For $G$ group and $H$ subgroup of finite index, prove that $N \subset H$ normal subgroup of $G$ of finite index exists
negative_passages.docid: 120642
negative_passages.text: Let $g : \Bbb N \times \Bbb N \to\Bbb N \times \Bbb N$ defined as $g(m,n) = (m + n,m - n)$