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I recently had my second abstract algebra exam returned to me and I missed this particular question (unfortunately I left the problem blank, just a bad test day). I've been trying to figure out the problem though and could use some help.

a) Does there exist a surjective homomphism $\phi: \mathbb{Z} \rightarrow S_3$? Explain.

What I'm thinking: Let $\phi: \mathbb{Z} \rightarrow S_3$ be a homomorphism.$\mathbb{Z}$ is a cyclic group and thus $\phi(\mathbb{Z})$ is cyclic. However, $S_3$ is not cyclic. Thus $\phi(\mathbb{Z}) \neq S_3$ and $\phi$ cannot be surjective.

b) Assume that there exist homomorphisms $\phi: \mathbb{Z} \rightarrow S_3$ where Ker$(\phi) \neq \mathbb{Z}$. What are the possible kernels? Explain.

What I'm thinking: Since $1$ generates $\mathbb{Z}$, if $1\in$ Ker$(\phi)$, then $\mathbb{Z}=$Ker$(\phi)$. So could Ker$(\phi)$ = $n\mathbb{Z}$ for $n \in \mathbb{Z}, \; n\neq1$?

c)List at least two non-trivial homomorphisms $\phi: \mathbb{Z} \rightarrow S_3$ whose kernals are not $\mathbb{Z}$.

$\phi_1(n) = (1\; 2)^n$

$\phi_2(n) = (1\; 3)^n$

Any input is much appreciated!

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a) is fine. But for b) Note that $n\in\ker \phi$ if $g^n=1$ for $g:=\phi(1)$. As such $g$ can only have orders $2$ or $3$ (or of course $1$) in $S_3$, only $2\mathbb Z$ and $3\mathbb Z$ (and of course $\mathbb Z$ that was excluded) are possible kernels.

c) is fine - though in the light of b) you were probably supposed to take $n\mapsto (1\,2\,3)^n$ as one of the examples.

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  • $\begingroup$ Can you briefly expand on part b? What does $g:=\phi(1)$ mean? $\endgroup$ – Zachary Luety Dec 4 '13 at 16:03
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The nice thing about the group $\mathbb{Z}$ is that it is the free group generated by one element. That is, you can map a generator (1 or -1) of $\mathbb{Z}$ to any element and that defines a (unique) group homomorphism. One way of seeing the existence of such a homomorphism is as follows:

Let $G$ be a group and $g \in G$. Then the subgroup $\langle g \rangle$ of $G$ is cyclic and hence is isomorphic to a quotient of $\mathbb{Z}$. This gives a surjection $\mathbb{Z} \twoheadrightarrow \langle g \rangle$ which sends $1$ to $g$. Composing this with the inclusion $\langle g \rangle \hookrightarrow G$ yields a group homomorphism from $\mathbb{Z}$ to $G$ that sends $1$ to $g$.

The homomorphism $\mathbb{Z} \rightarrow G$ with $1 \mapsto g$ has zero kernel if and only if $g$ has infinite order. Otherwise the kernel is $n \mathbb{Z}$ where $n$ is the order of $g$. So the nonzero kernels for the group homomorphisms from $\mathbb{Z} \rightarrow G$ are in one-to-one correspondence with the orders of the non-identity elements in $G$.

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