Homomorphisms from $S_{5}$ to $\mathbb{Z}_{12}$ , what does it mean the ID subgroup? I'm trying to find the number of homomorphisms from $S_{5}$ to $\mathbb{Z}_{12}$ , meaning : 
$S_{5}$ $\longrightarrow$ $\mathbb{Z}_{12}$
I'm using the 1st Isomorphism theorem .
$G/Ker(f)≅Im(f)$
So , we can have the following subgroups for Ker :


*

*Id 

*$A_{5}$

*$S_{5}$
For the first : $120/1≅Im(f)=120$ but by Lagrange theorem $|Im(f)|  | |\mathbb{Z}_{12}| $ , but 120 doesn't divide 12 , then this can't be .
Question : what does it mean the ID subgroup ? why its size is 1 ?  
Regards 

I suspect what is meant by "the ID subgroup" (which is terrible notation, by the way) is the identity subgroup, ie $\{e\}$, where $e$ represents the identity element of $S_5$.
In general, if $G$ is any group with identity element $e_G$, then $\{e_G\}$ is a normal subgroup of $G$.
