- #1
mathmari
Gold Member
MHB
- 5,049
- 7
Hey!
I want to show that $N\{1, (12)(34), (13)(24), (14)(23)\}$ is a normal subgroup of $S_4$ that is contained in $A_4$ and that satisfies $S_4/N\cong S_3$ and $A_4/N\cong Z_3$. Let $\sigma\in S_4$.
We have the following:
$$\sigma 1 \sigma^{-1}=\sigma (1) \\ \sigma (1 2)(3 4) \sigma^{-1}=\left (\sigma (1) \sigma (2)\right ) \left (\sigma(3) \sigma (4)\right ) \\ \sigma (13)(24) \sigma^{-1}=\left (\sigma (1) \sigma (3)\right )\left (\sigma (2)\sigma (4)\right ) \\ \sigma (14)(23) \sigma^{-1}=\left (\sigma (1)\sigma (4)\right )\left (\sigma (2)\sigma (3)\right )$$
Right? (Wondering)
But how could we show that these are elements of $N$ ? (Wondering)
I want to show that $N\{1, (12)(34), (13)(24), (14)(23)\}$ is a normal subgroup of $S_4$ that is contained in $A_4$ and that satisfies $S_4/N\cong S_3$ and $A_4/N\cong Z_3$. Let $\sigma\in S_4$.
We have the following:
$$\sigma 1 \sigma^{-1}=\sigma (1) \\ \sigma (1 2)(3 4) \sigma^{-1}=\left (\sigma (1) \sigma (2)\right ) \left (\sigma(3) \sigma (4)\right ) \\ \sigma (13)(24) \sigma^{-1}=\left (\sigma (1) \sigma (3)\right )\left (\sigma (2)\sigma (4)\right ) \\ \sigma (14)(23) \sigma^{-1}=\left (\sigma (1)\sigma (4)\right )\left (\sigma (2)\sigma (3)\right )$$
Right? (Wondering)
But how could we show that these are elements of $N$ ? (Wondering)