So I am looking to determine whether the following are surjective homomorphisms, and if so are they also isomorphisms by computing kernels:
I am completely new to the topic of kernels in group theory so I apologize if this doesn't make much sense. I really need help understanding these questions.
1) $G = \mathbb R, H = ((0,∞),×), f(x) = e^x$,
2) $G = H = \mathbb R, f(x) = 2x$
3) $G = (\mathbb C/{0} ,×), H = S^1, f(z) = \frac{z}{|z|}$
4) $G = O(n), H = \mathbb Z_2, f(A) = \det(A)$
5) $G = H = ((0,\infty),×), f(x) = x^2$
My attemp:t
1) We have $e^{x+y}=e^x*e^y=e^xe^y$ so $e^x$ is a homomorphism,
and the kernel $e=({x \in \mathbb R : e^x = e_{(0, \infty)}=1})=({0})$.
So $e^x$ is injective and hence an isomorphism.
2) $f(xy)=2(xy)$ which is not equal to $2x*2y$.
Hence it is not a homomorphism.
3) Really unsure how this works.... $f(zx)=\frac{zx}{|zx|}?$
4) $f(AB)=\det(AB)=\det(A)\det(B)$. Hence it is a homomorphism....
I'm not sure how to compute the kernel.
5) $f(xy)=(xy)^2=x^2y^2=f(x)*f(y)$. Hence this is a homomorphism;
the kernel would be $({x \in (0,\infty)|x^2 = (0,\infty)^2=1})$ (this doesn't look like it makes any sense).