Question

the number of group isomorphic from s3 to z/6z

Answer 1

To restate, you're asking for the number of isomorphisms \(S_3 \longrightarrow \mathbb{Z}/6\mathbb{Z}\) Where \(S_3\) is the permutation group of 3 elements correct?

Answer 2

yes

Answer 3

Let's think about this a little. What's one of the key properties that \(\mathbb{Z}/6\mathbb{Z}\) has?

Answer 4

is it a cyclic group
z/nz is isomorphic to nz

Answer 5

Both important properties, but in this case we know that it's also abelian.

Answer 6

As for \(S_3\), it is not abelian. Thus, the two groups are not isomorphic.

Answer 7

that means no of isomorphisms is zero

Answer 8

yes.

Answer 9

thanks will think about it bye