Suppose that $G$ is a finite group that can be generated from $n$ elements $g_1,...,g_n\in G$.
Now, if $S\subseteq G$ is another set of generators of $G$, can I always find a subset $S'\subseteq S$ of size $\le n$ that still generates $G$?
2026-03-27 13:10:38.1774617038
Does every set of generators of a finite group contain a minimal set of generators?
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No.
For example, $S_4$ is generated by $(1,2)$ and $(1,2,3,4)$, so it is $2$-generated. However, no subset of size $2$ in $\{(1,2),(2,3),(3,4)\}$ generates $S_4$.