How to prove that $|HK| = \dfrac{|H| \; |K|}{|H \cap K|}$?
89190
Why is it true that $|AB:A|=|B:A\cap B|$ even if $A$ is not normal in $AB$? (Second Isomorphism Theorem)
33524
Prove the following $A \setminus (B\setminus C) = (A\setminus B)\cap (A \cup C)$
147888
Show that $M=\bigcap_{\mathfrak{p}\in\operatorname{Spec}(R)}M_\mathfrak{p}=\bigcap_{\mathfrak{m}\in\text{Max}(R)}M_\mathfrak{m}$ for certain $M$.
15312
Show that $R \cap R^{−1}$ is an equivalence relation on $X$?
140446
Prove that $A \smallsetminus (A \smallsetminus B) = A \cap B$
94571
How can we prove that $N_G(H)=H$ implies $N_{f(G)}(f(H))=f(H)$
120985
Prove that $\mathrm{Res}[f/g,z_0]=\frac{f(z_0)}{g'(z_0)}$
65873
A proof for Atiyah-Macdonald Exercise I.21.iii
30770
Prove that $C\left(\bigcup_{\alpha\in I} A_\alpha\right)=\bigcap_{\alpha\in I}C\left(A_\alpha\right)$
26302
Prove that $(X \cap Y) \cup Z = (X \cup Z) \cap (Y \cup Z)$
45179
When does the isomorphism $G\simeq \ker(\phi)\times \operatorname{im}(\phi)$? hold?
48291
Proof that $\mathrm{ker}(f^*) = \mathrm{im}(f)^{\perp} $
1251
If $[G:K]$ is finite, then $[H:H \cap K] = [G:K]$ iff $G = HK$ (Hungerford Proposition 4.8, Proof)
35800
If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
89659
$H_1 ,H_2 \unlhd \, G$ with $H_1 \cap H_2 = \{1_G\} $. Prove every two elements in $H_1, H_2$ commute
156803
Proving a corollary to Schwarz lemma
2174
Proving that HK is a subgroup when K is normal