query: To Show that all ﬁbers of this map has exactly $|H \cap K|$ elements and $|HK| = \frac{|H|\times|K|}{|H\cap K|}$
positive_passages.docid: 5042
positive_passages.text: Order of a product of subgroups. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$.
negative_passages.docid: 120035
negative_passages.text: Pattern in Isomorphisms between $\mathbb Z_m \times \mathbb Z_n$ and $\mathbb Z_{mn}$
negative_passages.docid: 73174
negative_passages.text: Prove $KN\cong K\times N$ if $K\cap N=\{e\}$
negative_passages.docid: 144502
negative_passages.text: Showing that $A\cap B = A\setminus (A\setminus B)$
negative_passages.docid: 86717
negative_passages.text: Completeness of $\langle \mathscr{C} [0, 1], \| \cdot \|_1 \rangle$
negative_passages.docid: 114916
negative_passages.text: $\mathbb Z_{mn}$ isomorphic to $\mathbb Z_m\times\mathbb Z_n$ whenever $m$ and $n$ are coprime
negative_passages.docid: 89659
negative_passages.text: $H_1 ,H_2 \unlhd \, G$ with $H_1 \cap H_2 = \{1_G\} $. Prove every two elements in $H_1, H_2$ commute
negative_passages.docid: 62711
negative_passages.text: Canonical map $R/(I\cap J)\rightarrow R/I\times _{R/(I+J)} R/J$ is an isomorphism
negative_passages.docid: 111129
negative_passages.text: Show that $[K:F]_s = [K:L]_s [L:F]_s$ and $[K:F]_i = [K:L]_i [L:F]_i$.
negative_passages.docid: 38946
negative_passages.text: Show that $\{\cup_{n\in K} (n, n+1]: K \subset \mathbb{Z}\}$ is a $\sigma$-algebra on $\mathbb{R}$
negative_passages.docid: 64527
negative_passages.text: Show that an operation of elements in a finite group contains each element exactly once
negative_passages.docid: 49242
negative_passages.text: stab(S) is isomorphic to $ S_k \times S_{n-k} $
negative_passages.docid: 128256
negative_passages.text: How to prove $H_1(\mathbb{C}/\Lambda, \mathbb{Z})$ to the lattice $\Lambda$ ,$\gamma \mapsto \int_\gamma dz$ is well defined?
negative_passages.docid: 80876
negative_passages.text: Proof of $A\cap(B\cup C) = (A\cap B)\cup(A\cap C)$
negative_passages.docid: 53475
negative_passages.text: Let $H$ have order $m$ and $K$ have order $n$, where $m$ and $n$ are relatively prime. Then $H \cap K=\{e\}$
negative_passages.docid: 154837
negative_passages.text: mapping properties of $(1−z)^i$
negative_passages.docid: 52105
negative_passages.text: Prove that $HK$ is a subgroup iff $HK=KH$.