Let $G$ be a group.
Let $H,K$ be subgroups of $G$.
How do I prove that $[G:H\cap K]\leq [G:H][G:K]$?
Let's not assume any index is finite.
Then, still the result holds?
If so how do I prove it?
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I got it.
Let $A$ be the set of cosets of $H\cap K$
Let $B$ be the set of cosets of $H$
Let $C$ be the set of cosets of $K$
Define $\phi:A\rightarrow B\times C : g(H\cap K)\mapsto (gH,gK)$.
Note that $\phi$ is well defined and also it's trivial that it is injective
Thus, $|A|\leq |B||C|$.
However I'm really wondering whether my argument is correct or not, since above in the link, people are using group actions and isomorphism theorems and stuffs to prove this..