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I want to use the following Theorem:

If $H\leq U\leq G$ are (maybe) infinite groups. And $|U:H|<\infty$, $|G:U|<\infty$. Then

$|G:H|=|G:U|\cdot |U:H|$.

I think, i could proof it, but i don't want to proof it. I only need a reference.

Thanks for help.

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2 Answers 2

This is in Robinson's book A Course in the Theory of Groups. It is on page $11$ ($1.3.5$).

It is perhaps worth saying that I did this in my introductory groups course, which implies that it is probably in every group theory text which covers the basics. (Of course, Robinson's text is pitched at graduate level and above so it covers the basics pretty quickly, but it does cover them...what I am trying to say is that if you have Dummit and Foe, or Fuchs Algebra, or any other "standard" text then it will probably be in there. Robinson is my "go-to" book, and it is in there, so it is probably in everyone else's "go-to" books.)

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Thank you very much. –  Peter Apr 9 '13 at 9:46

If online references are ok (I'm nowhere near my books right now), then there is this

http://groupprops.subwiki.org/wiki/Index_is_multiplicative

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