1
$\begingroup$

What is a (simple) example of a group $G$ with an endomorphism $G\rightarrow G$ That is surjective, but not injective?

$\endgroup$
  • $\begingroup$ What could I have tried? I'm looking for an example, either I know one, or I don't $\endgroup$ – JimmyP Mar 23 '15 at 18:32
  • 3
    $\begingroup$ Yes, but it isn't as though the only options are "I know an example immediately" and "such an example will never occur to me no matter how long I think about it". What properties of $G$ most hold? For instance, could $G$ be finite? $\endgroup$ – BaronVT Mar 23 '15 at 18:39
2
$\begingroup$

Let $G= H^\infty$ for some non-trivial group $H$ (like $\Bbb Z$ or $\Bbb Z_2$), and let $\phi:G\to G$ be given by. $$ \phi(h_1,h_2,h_3,\ldots )=(h_2,h_3,\ldots) $$

$\endgroup$
  • $\begingroup$ Why is this surjective? $\endgroup$ – JimmyP Mar 23 '15 at 18:46
  • $\begingroup$ @JimmyP for any $(h_2,h_3,\dots)\in G$ there is an element $u\in G$ such that $\phi(u)=(h_2,h_3,\dots)$. E.g. $u=(e,h_2,h_3,\dots)$ where $e$ is the identity of $H$. $\endgroup$ – drhab Mar 23 '15 at 19:04
  • $\begingroup$ @drhab but isnt it also injective in that case? $\endgroup$ – JimmyP Mar 23 '15 at 19:07
  • $\begingroup$ Not if $H$ is not trivial. If $x\in H$ with $x\neq e$ then you can also take $v=(x,h_2,h_3,\dots)\neq(e,h_2,h_3,\dots)=u$. $\endgroup$ – drhab Mar 23 '15 at 19:09
  • $\begingroup$ @drhab I get it now, thanks! $\endgroup$ – JimmyP Mar 23 '15 at 19:11
2
$\begingroup$

There are no examples with $G$ finite. So you need $G$ to be infinite.

Consider the set of all polynomials with rational coefficients as an additive group.

Then the derivation map $p\mapsto p'$ is a surjective, but not injective, endomorphism.

Or consider the set of complex numbers of modulus $1$ as a multiplicative group.

Then the square map $z\mapsto z^2$ is a surjective, but not injective, endomorphism.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.