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I'm reading Wagon's book "The Banach-Tarski paradox". At page $149$, the author talks about the "direct union of a directed system of amenabile groups" without defining it.

Q1: does direct union simply mean union? If this is the case, why does he use the adjective "direct"? Or is direct union the same as direct limit?

Q2: the same author states that "any group is the direct union of its finitely generated subgroups". It seems clear to me that finitely generated subgroups form a direct system (right?) and it seems almost trivial that the union of all finitely generated is a subgroup that, actually, equals the group itself (right?).

Thank you in advance for your help.

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  1. For Q1 this means "direct limit." I suspect that in his setting, each homomorphism $f_{ij}: G_i\to G_j$ in the directed system is injective, so it makes sense to identify groups $G_i$ with subgroups of one large group $G$ such that each $f_{ij}$ is the inclusion map $G_i< G_j$ and $G$ is the union of the subgroups $G_i$.

  2. For Q2, just disregard the word "direct," each group is simply the union of its finitely generated subgroups, which is quite obvious.

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To say that some $X$ is the direct union of a collection of some $Y_i$'s means that $X$ is the union of the $Y_i$'s and that the $Y_i$'s form a directed family (any two of them are included in one of them).

So "direct union of a directed system" is redundant. You could omit "direct" or "directed" (but not both) without changing the meaning.

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