Homomorphism from the group of permutation to the group of non zero complex numbers.

Question

The question I am asking is related to this one.

My question is what if the codomain is interchanged with the domain. i.e is there any nontrivial homomorphism between the permutation group (the group of even permutations) of $n(n \ge 2)$ elements and the the group of nonzero complex numbers ?

Thank you.

Answer 1

The homomorphisms from a group $G$ to an abelian group $A$ are in one-to-one correspondence with homomorphisms to $A$ from the abelianization $G^{\mathrm{ab}}$ of $G$.

We have (according to wikipedia):

• $S_1^{\mathrm{ab}} = 0$
• $S_n^{\mathrm{ab}} = \mathbb{Z} / 2 \mathbb{Z}$ for $n \geq 2$
• $A_3^{\mathrm{ab}} \cong A_4^{\mathrm{ab}} \cong \mathbb{Z} / 3 \mathbb{Z}$
• $A_n^{\mathrm{ab}} = 0$ for $n \neq 3,4$

and its easy to find nontrivial homomorphisms from nontrivial cyclic groups to $\mathbb{C}^\times$. For finite ones, think of the roots of unity.