0
$\begingroup$

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.

$\endgroup$
  • 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
0
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.