compute kernel for S3-Z2

Question

compute kernel for S3->Z2

anonymous · Answer

$$\phi:S_{3}\rightarrow Z _{2}$$

anonymous · Answer

are  you given the map?

anonymous · Answer

kernel will have order 3, so maybe all the rotations get sent to zero and the flips to one?

kinggeorge · Answer

I'm pretty sure (assuming this is a homomorphism) that any element of order 2 in$S_3$ must be sent to 1, and all others are sent to $0$.

anonymous · Answer

that sound right.  in fact it has to be right

anonymous · Answer

i am not sure what you are naming your elements of $S_3$ but the rotations will get sent to zero, flips to 1.

kinggeorge · Answer

Unless of course it's not a homomorphism. If that's the case anything goes.

anonymous · Answer

how we want to know the mapping?i mean how you know any element of order 2 in S3 must be sent to 1 and others sent to 0??

kinggeorge · Answer

$\mathbb{Z}_2 =\{0, 1\}$ and 1 has order two. Thus, at least one element of $S_3$ must  be sent to 1, and it must have order 2. By that, we know that every element in $S_3$ that has order not equal to 1 is sent to 0. With a little bit of work, you can show that the element that was sent to 1 can generate the other elements of $S_3$, and using the properties of a homomorphism, you can show that they are all sent to 1.

anonymous · Answer

means that the kernel $$\phi$$ is the element in $$S_{3}$$ that sent to 0??

kinggeorge · Answer

That is what the kernel would be.

anonymous · Answer

yeah2...i try first..

anonymous · Answer

why you say that element that sent to 1 can generates other element of S3?the order is 2..