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Does any infinite group contain a minimal set of generators?

This is not true for semigroups. But for groups?

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    Presumably you mean "Does every infinite group contain a minimal set of generators?". The word "any" is often ambiguous.2011-10-15

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The answer is "no". For example, consider the Prüfer $p$-group $\mathbf{Z}_{p^{\infty}}$. A subset $X$ of $\mathbf{Z}_{p^{\infty}}$ generates if and only if it contains elements of arbitrarily large order. In particular, if $X$ is any generating set, then you can always remove a finite number of elements from $X$ and still have a generating set, so $\mathbf{Z}_{p^{\infty}}$ has no minimal generating set.

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    The additive group of rational numbers is another example.2011-10-15