1. ## Homomorphism and Kernels

Struggling with these questions...help needed ASAP please!

Q1. Define f: R \ {0} --> R \ {0} by f(x) = x^2 (the group operation is multiplication). Prove that f is a homomorphism and give the kernel. Is f an isomorphism?
Q2. Prove that {2^k | k E Z} is a subgroup of (R \ {0}, .).

2. Originally Posted by wik_chick88
Struggling with these questions...help needed ASAP please!

Q1. Define f: R \ {0} --> R \ {0} by f(x) = x^2 (the group operation is multiplication). Prove that f is a homomorphism and give the kernel. Is f an isomorphism?
to show it is a homomorphism you must show that $f(a*b) = f(a)*f(b)$ for all $a,b \in \mathbb{R} \backslash \{ 0 \}$

can you show this? try it and tell us what you get.

the kernel of $f$, $\mbox{ker}f$, is the set of all $x \in \mathbb{R} \backslash \{ 0 \}$ such that $f(x) = 1$. (why 1? because it is the identity element with respect to multiplication on the reals). So, what numbers are in this set?

Here is what an isomorphism is: in a nutshell, it is a homomorphism that is one-to-one and onto.

so, is $f(x) = x^2$ one-to-one and onto?

3. Originally Posted by wik_chick88
Q2. Prove that {2^k | k E Z} is a subgroup of (R \ {0}, .).
you must show three things.

(1) You must show that the set is nonempty (it is always good to check that the identity of the group, that is, $( \mathbb{R} \backslash \{ 0 \}, \cdot )$ is in the subgroup)

(2) the set is closed under multiplication. meaning, that if you multiply any two elements in the set, the result is also in the set.

(3) you must show that for each element in the set, its (multiplicative) inverse is also in the set.

can you continue? take it one step at a time