Question

If `f` is a group homomorphism from `S_3` into `ZZ_6`, then what is the order of `f(S_3)` ?

Answer 1

`1` or `2`

Explanation:

Since `ZZ_6` contains an element of order `6`, but `S_3` does not, there is no group isomorphism. So the image of `S_3` in `Z_6` is of order strictly dividing `6`.

Consider the image of the transposition `((1, 2))` under `f`. It must either be an element of order `1` or of order `2`. It cannot have order `3`.

So we either have `f(((1,2))) = hat(0)` or `f(((1,2))) = hat(3)`

Note that if `((m,n)) in S_3` then there is some permutation `p` in `S_3` such that: `p^(-1) ((1, 2)) p = ((m, n))`. Hence:

• If `f(((1,2))) = hat(0)` then `f(((m,n))) = hat(0)` for all transpositions in `S_3` and hence `f(S_3) = { hat(0) }`

• If `f(((1, 2))) = hat(3)` then `f(((m,n))) = hat(3)` for all transpositions in `S_3` and hence `f(S_3) = { hat(0), hat(3) }`