1

Let $H$ and $K$ be subgroups of a group $G$. Then is the following true ?

If $[G : H] < ∞$ and $[G : K] < ∞$, then $[G : H ∩ K] < ∞$.

I think this's false because there's still a case that $H ∩ K$ could be empty. But the textbook requires to prove this statement. So I am a bit confusing on this point.. any help?

Teddy38
  • 3,309
snapper
  • 589
  • 1
    The intersection of two subgroups can't be empty. – Gerry Myerson Apr 19 '18 at 07:31
  • @GerryMyerson Because of $e$? – snapper Apr 19 '18 at 07:33
  • 1
    Right. See also https://math.stackexchange.com/questions/177539/my-approach-to-does-the-intersection-of-two-finite-index-subgroups-have-finite and https://math.stackexchange.com/questions/128538/does-the-intersection-of-two-finite-index-subgroups-have-finite-index? and https://math.stackexchange.com/questions/886868/prove-that-intersection-of-finite-index-subgroups-has-finite-index – Gerry Myerson Apr 19 '18 at 07:34
  • @GerryMyerson better be marked as duplicate – snapper Apr 19 '18 at 07:39

0 Answers0