[Math] Group Homomorphism Questions (the attempts shown)

Question

(a) Let $p$ be a prime. Determine the number of homomorphisms from $\Bbb Z_p \oplus \Bbb Z_p$ into $\Bbb Z_p$.

Attempt: Suppose $\Psi:Z_p \oplus Z_p \rightarrow Z_p$ is an into homomorphism.

Then $|Ker~\Psi|= p^2/x $ when $x = |\Psi (Z_p \oplus Z_p)| < p$

The only $p^2/x $ is an integer when $x=1$. Hence, there exists only the trivial homomorphism when $|\Psi (Z_p \oplus Z_p)| < p$ .

IS this correct?

Now, if we are asked the number of onto homomorphisms from $Z_p \oplus Z_p$ onto $Z_p$?

Attempt: If $\Psi:Z_p \oplus Z_p \rightarrow Z_p$ is an onto homomorphism, then $|Ker~\Psi|=p$

Since $Z_p$ is a cyclic (hence, normal) group, thus, $\Psi^{-1}(Z_p)$ is also a normal subgroup of $Z_p \oplus Z_p$ . (Anyways, every subgroup of an abelian group is normal)

I really am not able to move forward now.

(b) Determine all homomorphisms from $\Bbb Z$ onto $S_3$.

Attempt: $\Psi(1)$ can take $6$ values since $|S_3|=6$ . Hence, there are 6 possible homomorphisms ?

But, let $\Psi : Z \rightarrow S_3$ be a homomorphism. Since, the kernel of a homomorphism is a normal subgroup of the domain, and every subgroup of $Z$ is of the form $nZ \implies $ kernel of $Z$ must be of the form $nZ$ for some $n$. But $Z/nZ \approx Z_n$ and by the first isomorphism theorem : $Z/nZ$ should be isomorphic to $S_3$ but $S_3$ is not cyclic. So, how can this homomorphism hold true?

Now, we got to find all homomorphisms from $Z \rightarrow S_3$

Attempt: Since, $|S_3|=6$ , this means since the divisors of $6$ are $2,3$, we need to find cyclic subgroups of orders $2$ and $3$ in $S_3$. Let them be $H_2$ and $H_3$ respectively.

Then $Z/2Z \approx H_2$ and $Z/3Z \approx H_3$

IS this approach correct?

(c) Suppose that the number of homomorphisms from $G \mapsto H$ is $n$. If $H$ is abelian, how many homomorphisms are there from $G \oplus G\cdot\cdot \cdot \oplus~ G ( s $ times) to $H$?

I have really No idea how to proceed in this.

I have posted this question earlier but the book which I am reading (Gallian) has not introduced any such concepts as given in the answer.

Your help will be really appreciated.
Thank you for your time and patience.

Answer 1

Hint: To classify/count homomorphisms $G_1\to G_2$ between finite groups, it is often enough to look at the image of the generators of $G_1$, which will determine the whole homomorphism.

Examples:

1. We want to find all homomorphism $\mathbb{Z}\to\mathbb{Z}_n$. The group $\mathbb{Z}$ is generated by the element $1$, and we have no relations (it is the free group on one element), thus $1$ can map to any element $x\in\mathbb{Z}_n$. This implies that we have exactly $n$ such homomorphisms, given by $k\mapsto nk$.
2. Now we look for homomorphisms $D_3\to\mathbb{Z}_3$. The group $D_3=\left<r,s|r^3,s^2,(rs)^2\right>$ is finitely presented and generated by the two elements $r,s$. Notice that $r$ must map to an element of order $3$ (so anything goes), and $s$ to an element of order $2$ (so we must have $s\mapsto 0$). You can easily check that we have $3$ possible homomorphisms, given by $s\mapsto 0$ and $r\mapsto x$ with $x=0,1,2$.