- #1

- 92

- 0

How many homomorphisms are there from [itex]S_5[/itex] to [itex]\mathbb{Z}_5[/itex]?

Well there is at least one, the trivial homomorphism, ie: every element of S_5 gets mapped to 0.

I have a feeling that this is the only homomorphism but am having trouble proving that no other homomorphism could exist. Any suggestions?

I know that every nonzero element of Z_5 has order of 5, and for a non-trivial homomorphism to exist, there needs to be some element of S_5 (not equal to the identity) that gets mapped to a nonzero element of Z_5.

Well there is at least one, the trivial homomorphism, ie: every element of S_5 gets mapped to 0.

I have a feeling that this is the only homomorphism but am having trouble proving that no other homomorphism could exist. Any suggestions?

I know that every nonzero element of Z_5 has order of 5, and for a non-trivial homomorphism to exist, there needs to be some element of S_5 (not equal to the identity) that gets mapped to a nonzero element of Z_5.

Last edited: