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We are given a Group $G$ and subgroups $X$ and $Y$. Prove that

$$|XY| |X\cap Y|= |X||Y|$$

I don't know how to proceed to prove this identity. Please help.

Thanks in advance.

Community
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Henry
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1 Answers1

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Don't forget that $XY$ need not be a subgroup of $G$. The direct product $X\times Y$ has a left action on $G$ via $(x,y)\bullet g=xgy^{-1}$ and $XY$ is the orbit of the identity element $e$. The stabiliser of $e$ is $\{(x,y)\in X\times Y: xey^{-1}=e\}=\{(x,x):x\in X\cap Y\}$. Now use the orbit-stabiliser theorem.

Angina Seng
  • 158,341
  • I don't know about the Orbit Stabilizer Theorem yet. I'm looking for an elementary combinatorial proof. – Henry Dec 26 '17 at 06:42
  • @Henry Group actions are well worth learning about. But you can ask yourself: when does $xy=x'y'$? – Angina Seng Dec 26 '17 at 06:44