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Abstract Algebra

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Surjective Group Homomorphism, Epimorphism, Onto

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Let $(G_1,\circ_1)$ be a group with composition law $\circ_1$ and $(G_2,\circ_2)$ a group with composition law $\circ_2$.

Let $f:G_1 \rightarrow G_2$ be a group homomorphism.

The group homomorphism $f$ is defined to be surjective if and only if:

for every $g_2\in G_2$ there is a $g_1\in G_1$ with $f(g_1)=g_2$

...a synonym for "surjective homomorphism" is "epimorphism", and we also say, ​"the homomorphism is onto".

Which of the following are true?

Select ALL that apply.

A

Let $({\rm UT}(2,{\mathbb R}), \times)= \left\{\begin{pmatrix}1&a\cr0&1\end{pmatrix}\mid a\in {\mathbb R}\right\}$ under matrix multiplication, and:

$f:({\rm UT}(2,{\mathbb R}), \times)\rightarrow ({\mathbb R}, +)$

...be given by:

$f\left(\begin{pmatrix}1&a\cr0&1\end{pmatrix}\right)=a$

...then $f$ is not a surjective homomorphism of groups.

B

Let $({\rm AT}(2,{\mathbb R}), +)= \left\{\begin{pmatrix}m&a\cr0&m\end{pmatrix}\mid m\in{\mathbb Z},\;a\in {\mathbb R}\right\}$ under matrix addition, and:

$f:({\rm AT}(2,{\mathbb R}), +)\rightarrow ({\mathbb R}, +)$

...be given by:

$f\left(\begin{pmatrix}m&a\cr0&m\end{pmatrix}\right)=a$

...then $f$ is a surjective homomorphism of groups.

C

Let $m\ge2$ be an integer and:

$f:({\mathbb Z}, +)\rightarrow (\{z\in{\mathbb C}\mid |z|=1\},\times)$

...be given by:

$f(a)=e^{2\pi ia/m}$

...then $f$ is a surjective homomorphism of groups.

D

Let $\theta$ be an irrational number and:

$f:({\mathbb Z}, +)\rightarrow (\{z\in{\mathbb C}\mid |z|=1\},\times)$

...be given by:

$f(a)=e^{2\pi i\theta a}$

...then $f$ is a surjective homomorphism of groups.

E

Let $\theta$ be an irrational number and:

$f:({\mathbb R}, +)\rightarrow (\{z\in{\mathbb C}\mid |z|=1\},\times)$

...be given by:

$f(a)=e^{2\pi i\theta a}$

...then $f$ is a surjective homomorphism of groups.