query: Let $H$ and $K$ be normal subgroups of $G$ such that $H \cap K = \langle e \rangle$ and $HK = G$. Prove that $G$ is isomorphic to $H \times K$.
positive_passages.docid: 151621
positive_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: 27507
negative_passages.text: Subgroups of Symmetric groups isomorphic to dihedral group
negative_passages.docid: 142413
negative_passages.text: Normal Subgroups of $SU(n)$
negative_passages.docid: 61379
negative_passages.text: Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$.
negative_passages.docid: 56245
negative_passages.text: Let H be a subgroup of a group G. Prove that the following statements are equivalent.
negative_passages.docid: 35353
negative_passages.text: Isomorphic quotient groups $\frac{G}{H} \cong \frac{G}{K}$ imply $H \cong K$?
negative_passages.docid: 46260
negative_passages.text: Prove that a group generated by two elements of order $2$, $x$ and $y$, is isomorphic to $D_{2n}$, where $n = |xy|.$
negative_passages.docid: 9140
negative_passages.text: Show $\operatorname{Aut}(C_2 \times C_2)$ is isomorphic to $S_3=D_6$
negative_passages.docid: 102979
negative_passages.text: If $[G:H]$ and $[G:K]$ are relatively prime, then $G=HK$
negative_passages.docid: 77221
negative_passages.text: Prove that $G$ has a subgroup isomorphic to $G/H$.
negative_passages.docid: 29445
negative_passages.text: Trying to prove that $H=\langle a,b:a^{3}=b^{3}=(ab)^{3}=1\rangle$ is a group of infinite order.
negative_passages.docid: 107511
negative_passages.text: If $G$ and $H$ are nonisomorphic group with same order then can we say that $Aut(G)$ is not isomorphic to $Aut(H)$?
negative_passages.docid: 93596
negative_passages.text: If $H\leq G$ and $[G:H]=2$, then $H$ is normal in $G$.
negative_passages.docid: 98765
negative_passages.text: Showing that $G/(H\cap K)\cong (G/H)\times (G/K)$
negative_passages.docid: 65744
negative_passages.text: How can I Prove that H=G or K=G
negative_passages.docid: 146028
negative_passages.text: $GL(n, \mathbb{C})$ is isomorphic to a subgroup of $GL(2n, \mathbb{R})$
negative_passages.docid: 23579
negative_passages.text: If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$