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I'd like to know where to find it since it's very used in the articles of Loday and others.

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Have you looked at Curtis' notes on simplicial homotopy theory? I think what you're looking for is in there. I believe the notes appeared in Advances in the 70's. –  Dan Ramras Mar 9 '12 at 5:11
Another suggestion, which is a comment not an answer because I don't really understand the material well enough to see if it really answers your question: Chapter 8 of Weibel's book on Homological Algebra. Although the main emphasis is on simplicial objects in abelian categories, the remarks and exercises do develop some of the parallel theory for simplicial groups –  Yemon Choi Mar 9 '12 at 6:37

2 Answers 2

Oh, come on! Prop. 17.4, p. 69, of my ancient but still current book ``Simplicial objects in algebraic topology'' proves that the homology groups of the Moore complex of a simplicial group $G$ are the homotopy groups (defined simplicially) of the Kan complex $G$. That G is a Kan complex is Thm 17.1. The homotopy groups of any Kan complex agree with the homotopy groups of its geometric realization (op cit, Thm 16.6). The cited book is still available from the University of Chicago Press (or Amazon of course).

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There seems to be a proof in Moore's notes on Algebraic Homotopy Theory. There's a copy up on my web site as the first set of links here: http://faculty.tcu.edu/gfriedman/notes/ The material you want seems to be the beginning of Chapter 2.

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Thanks for making these available. My copy disappeared years ago! –  Ronnie Brown Mar 9 '12 at 10:31

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