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Let $G$ and $H$ be two finitely generated groups, and let $W = G \wr H$ be the wreath product of $G$ and $H$. Show that $W$ is finitely generated. In class today, we were showed this and told that it was obvious. However, I do not see how it is obvious. How is this obvious?

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    @Arturo: Well, in the comments Bernard says that $H$ is finite, in which case it doesn't matter. But I don't know why he says that, when it isn't in the question and, provided he means the restricted wreath product, it isn't necessary.2012-04-13

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