I'm trying to calculate how many homomorphisms exists from $Z_{12}$ $\longrightarrow$ $S_{5}$ . Here are the options :
$(x x x x x)$ order 5
$(xxxx)$ order 4
$(xx) (xx) $ order 2 ???
$(xxx)(xx) $ lcm(2,3)=6 , hence order 6
$(xxx)$ order 3
$(xx)$ order 2
$id$ , meaning every element is mapped to itself , order 1
My questions are :
why is the order of $ (xx) (xx) $ is 4 , and not 2 ?
We need $Z_{12}$ , so all the above has order 12 in common (lcm) except for $(x x x x x)$ with order 5 . Then this means that I need to find the ones that divide with 5 and subtract it from $|S_{5}|$
Regards