Does any infinite group contain a minimal set of generators?
This is not true for semigroups. But for groups?
Does any infinite group contain a minimal set of generators?
This is not true for semigroups. But for groups?
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.