Let $S_n$ be the symmetric group of $n$ elements $\{ 1,2,...,n \}$ and $G$ be a subgroup of $S_n$ or order $n$. Denote the elements in $G$ by $\{ \sigma_1,...,\sigma_n \}$. Let the matrix $A=(\sigma_i(j))\in M_{n\times n}(\mathbb Z)$. Is the matrix $A$ invertible?
Invertibility of one matrix constructed by order n subgroup of symmetric group
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