Consider $$K=\begin{Bmatrix}\begin{pmatrix}123\\123\end{pmatrix},\begin{pmatrix}123\\231\end{pmatrix},\begin{pmatrix}123\\312\end{pmatrix}\end{Bmatrix}$$

(A) Show that K is a normal subgroup of S3.

(B) Show that S3/K is isomorphic to 2

This exercise if I am not mistaken is in the book of Abstrac Algebra: An Introduction by Thomas Hungerford my doubt about it is to get to point b how to prove if it is isomorphism to 2.

  • 1
    $\begingroup$ You could always note that $S_3 / K$ has order $2$ hence must be $\cong \mathbb{Z}_2$. $\endgroup$ – Zain Patel May 30 '17 at 18:40
  • $\begingroup$ Do you know what is the sign of a permutation? If you see this permutations are the even ones. $\endgroup$ – Phicar May 30 '17 at 18:40
  • $\begingroup$ What do you know about the definition of the quotient group? $\endgroup$ – Ethan Bolker May 30 '17 at 18:40
  • $\begingroup$ Sorry, I just graduated and I want to learn about abstract algebra before I enter college and I still do not get very well on permutation ...I am still practicing and learning more thoroughly, I hope to get at least the rings $\endgroup$ – marie May 30 '17 at 18:44

Hint: Consider the following function $$\varphi : S_3\longrightarrow \mathbb{Z}_2,$$ such that $\varphi (\sigma)=|\{(i,j):i<j \hspace{2mm}\wedge \sigma _i>\sigma _j\}|\pmod 2.$

Check that $\varphi (K)=0.$ Show that $\varphi(\sigma \circ\tau)=(\varphi(\sigma)+\varphi (\tau))\pmod 2.$
Use first theorem of isomorphism.

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