What exactly is $\phi(G)$ in a homomorphism?

Question

I'm learning about group homomorphisms and I'm confused about what the $\phi$ transformation is exactly.

If we have some group homomorphism $\phi : G\rightarrow H$ what exactly does $\phi(G)$ mean?

Thanks for your time.

Related, possibly helpful: https://math.stackexchange.com/questions/2039702/what-is-an-homomorphism-isomorphism-saying/2039715#2039715 — Ethan Bolker, Dec 05 '18 at 18:31
I found your analogy very useful. Thanks for the link and your time. — User3213651, Dec 05 '18 at 21:25

score 4 · Accepted Answer · answered Dec 05 '18 at 18:28

4

$\phi(G)$ is the image of $G$ under the mapping $\phi$, i.e. the set of all $\phi(x)$ for $x \in G$.

answered Dec 05 '18 at 18:28

Robert Israel

448,999

Awesome, exactly what I was looking for. Thank you! – User3213651 Dec 05 '18 at 18:30

score 1 · Answer 2 · answered Dec 05 '18 at 18:28

1

$\phi(G)$ denotes the image of the group $G$. This is actually a purely set-theoretic definition. Given a map of sets $f:X\to Y$, $f(X)$ denotes the image of $f$. That is, $f(X)\subseteq Y$ is $\{y\in Y: y=f(x)\:\text{for some}\:x\in X\}$.

answered Dec 05 '18 at 18:28

Alekos Robotis

26,184

Great thank you so much for your detailed answer! – User3213651 Dec 05 '18 at 18:29

What exactly is $\phi(G)$ in a homomorphism?

2 Answers2