If #f# is a group homomorphism from #S_3# into #ZZ_6#, then what is the order of #f(S_3)# ?
1 Answer
Sep 9, 2017
Explanation:
Since
Consider the image of the transposition
So we either have
Note that if

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) }#