Find the Number of Group Homomorphisms from $\mathbb{Z}/10\mathbb{Z}$ to $A_4$.

Solution 1:

Yes, your reasoning is correct. Yet another way to tackle the problem is by the fundamental homomorphism theorem. Here we are looking for the (number of) possible subgroups of $A_4$ isomorphic to quotients of $\Bbb Z_{10}$; the (normal) subgroups of $\Bbb Z_{10}$ are: $\{0\}$, $\{0,5\}\cong \Bbb Z_2$, $H:=\{0,2,4,6,8\}\cong \Bbb Z_5$ and $\Bbb Z_{10}$; for order reasons (Lagrange theorem), the only options are $\Bbb Z_{10}/\Bbb Z_{10}\cong\Bbb Z_1$ and $\Bbb Z_{10}/H\cong\Bbb Z_2$. Therefore, if we call $\phi$ such a homomorphism, we can have either $\phi(\Bbb Z_{10})=\{()\}$ (the trivial case) or any among $\phi(\Bbb Z_{10})=\{(),(12)(34)\}$, $\phi(\Bbb Z_{10})=\{(),(13)(24)\}$, $\phi(\Bbb Z_{10})=\{(),(14)(23)\}$, so $4$ altogether.