Is the symmetric group $S_n$ a normal subgroup of the general linear group $GL(n,\mathbb{R})$? We regard $\sigma\in S_n$ acts on $\mathbb{R}^n$ by permuting the coordinates $\sigma(x_1,\cdots,x_n)=(x_{\sigma(1)},\cdots,x_{\sigma(n)})$.
2026-03-27 22:04:56.1774649096
symmetric group as a subgroup of general linear group
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No, $$\pmatrix{1 &1 \\ 0 &1}\pmatrix{0 &1 \\ 1 &0}\pmatrix{1 &1 \\ 0 &1}^{-1} = \pmatrix{1 &0 \\ 1 &-1}$$ is not a permutation matrix.