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Here is my definition for a grouplike space or an $H-$group from "Introduction to homotopy theory" by Martin Arkowitz:

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And here is a question I found here:

Every loop space $(\Omega Y,w_0)$ has the structure of an $H$-group. that is answering my question which is Why are loop spaces $\Omega Y$ examples of grouplike spaces ?, but unfortunately I do not understand the proof in that question, so I need a proof in terms of my definition. Could anyone show me that proof, please?

Also, if anyone could explain to me how is my definition is the same as the definition given in the link above (if that is corect) I would really appreciate that.

Eric Wofsey
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    Are you sure that your definition is different from the one used in the linked question? As far as I can tell, the question you linked only asks for a certain part of the proof (concerning part 1. in your definition) but the definition should be just the one you use. – Matthias Klupsch Sep 17 '20 at 14:48
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    Loop spaces are studied in detail in $\S$ 2.3 of Arkowitz. You'll find everything you need in that section. – Tyrone Sep 17 '20 at 15:02
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    The basic idea is all the arguments you make to prove the fundamental group is a group can be thought of on the level of spaces as proving that $\Omega Y$ is an H-group. –  Sep 17 '20 at 19:25
  • @MatthiasKlupsch Ok. I will reread the linked question and see, maybe I am not understanding it. –  Sep 17 '20 at 20:48
  • @JustinYoung are you speaking about the fundamental group $\pi_{1}(X)$ of in topology? –  Sep 17 '20 at 20:50
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    @Smart20 Yes. Observe that $\pi_0(\Omega Y) = \pi_1(Y)$. –  Sep 18 '20 at 15:43

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