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Let $G$ be a group. A set $S \subseteq G$ is called a generating set for $G$ if every element of $G$ can be written as a product of elements of $S\cup S^{-1}$. Call $S$ a minimal generating set for $G$ if there is no proper subset of $S$ which is a generating set for $G$. Different minimal generating sets can have different cardinalities. For example, $\{1\}$ and $\{2, 3\}$ are both minimal generating sets for $\mathbb{Z}$.

Is there an example of a finite group $G$ with two minimal generating sets with different cardinalities?

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up vote 5 down vote accepted

Recall that the symmetric group $S_n$ is generated by an order $2$ element and an order $n$ element. But also $\{(12),(13),\dots,(1n)\}$ generates all of the order $2$ elements, and any element of $S_n$ can be represented as a product of order $2$ elements.

For an abelian example, consider $\mathbb Z/15\mathbb Z$; both $\{1+15\mathbb Z\}$ and $\{3+15\mathbb Z,5+15\mathbb Z\}$ are minimal generating sets.

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