Why is it true that $|AB:A|=|B:A\cap B|$ even if $A$ is not normal in $AB$? (Second Isomorphism Theorem)
5042
Order of a product of subgroups. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$.
49143
Showing $A \subset B \iff A\cap B=A$
13012
Why can a = b imply 2 = 1?
23579
If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$
80202
Why is $[0, 1] \cap \mathbb{Q}$ not compact in $\mathbb{Q}$?
138069
Proof that $\bigcap_{a\in A} \bigg(\, \bigcup_{b\in B} F_{a,b} \, \bigg) = \bigcup_{f\in ^AB} \bigg(\, \bigcap_{a \in A} F_{a,f(a)}\,\bigg) $
100303
A necessary and sufficient condition for $A\cap(B+C)=(A\cap B)+C$ to hold for submodules
37061
The order of $ab$ when $a,b$ commute
143979
How can I prove that $|A| + |B| = |A\cup B| + |A\cap B|$?
77361
How to prove $|H|[K:K \cap (g^{-1}Hg)]=|K|[H:H \cap (gKg^{-1})]$?
93596
If $H\leq G$ and $[G:H]=2$, then $H$ is normal in $G$.
10107
Prove $A\cup (A\cap B) = A$
140503
What properties do natural isomorphisms between functors preserve?
12927
$A^2+B^2=AB$ and $BA-AB$ is non-singular
123890
Show $A\setminus (A\cap B)=(A\cup B)\setminus B$
88982
Prove that for any two sets $A$ and $B$, $A\notin B$ or $B\notin A$
147253
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 $[G:N]=p$