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Index of Intersection of Subgroups with Finite Index

  1. Jan 24, 2017 #1
    1. The problem statement, all variables and given/known data
    Suppose that ##H## and ##K## are subgroups of finite index in the (possibly infinite) group ##G## with ##|G : H|m## and ##|G:K|=n##. Prove that ##lcm(m,n) \le |G : H \cap K | < mn##.

    2. Relevant equations


    3. The attempt at a solution

    I was able to get the upper bound on ##|G : H \cap K|##, but am having difficulty showing that the lower bound is ##lcm(m,n)##. I tried showing that ##m## and ##n## both divide ##|G : H \cap K |##, but I couldn't get anywhere. I could use some hints!
     
  2. jcsd
  3. Jan 26, 2017 #2

    fresh_42

    Staff: Mentor

    I think you can use the theorem of Lagrange to get ##|G : H \cap K|=|G : K| \cdot |K : H \cap K|## so ##n\,\vert \,|G : H \cap K|## and similar ##m\,\vert \,|G : H \cap K|##.
     
  4. Jan 27, 2017 #3
    Unfortunately, Lagrange's theorem only applies to finite groups.
     
  5. Jan 27, 2017 #4

    fresh_42

    Staff: Mentor

    I haven't checked the proof, but on the Wiki page it has been first stated (## |G|=|G : H| \cdot |H| ##) and then appended "Especially for ##|G|<\infty \; \ldots ##" so I assumed that finiteness of ##G## isn't really required.
     
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