Multiplicative groups 
  
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*Consider the multiplicative groups $\def\R{{\mathbb R}}(\R^\ast,\cdot)$ and $(\R^+, \cdot)$.  Show that $\R^\ast \cong\R^+\times\Bbb Z_2$.
  
*Consider the  multiplicative group $\def\Q{{\mathbb Q}}(\Q^+,\cdot)$. List the elements of the cyclic subgroup $H=\langle4\rangle$. Is $H$ normal?  List the elements of the cosets $\frac13H$ and $2H$. Describe the quotient $\Q^+/H$ by picking a representative for each coset of $H$.  In the quotient $\Q^+/H$, what is the order of $\frac13H$? What is the order of $2H$?  What is $(2H)\Bigl(\frac23H\Bigr)$?
  

For $(1)$, I thought if $a$ is in $\mathbb{R}^*$, then we can find function $F$ such that $F(a) = a \times 1$ if $a>0$, 
and $F(a) = -a \times 0$ if $a<0$. Maybe this function shows they are isomorphic? Is this right?  
For $(2)$, how $H=\langle 4\rangle$ is cyclic subgroup? Is $H$ all $\mathbb{Q}*$, since if $a$ is in $\mathbb{Q}*$, we can always find $\frac14a$ in $\mathbb{Q}*$ or is $H$ just $\{4,8,12,16,\dots\}$ or $\{\dots,-8,-4,0,4,8.12,\dots\}$ ? I'm not sure what $H$ is.
To describe the quotient $Q^+$/H, maybe it is a set of aH such that 
$0 < a\leq 1$ and 
$a= \dfrac{x}{y}$ such that x and y are integers and $gcd(x,y) = 1$. 
Thank you!
(Original image of questions: http://i.stack.imgur.com/wRRUJ.png)
 A: Yes the first one is correct. In fact, using the map $a\to |a|$ would be effective here. There is another point I 'd like to note about the first one. That is, if we set $U=\{z\in\mathbb C\mid |z|=1\}$ then we have also $$\mathbb C/U\cong\mathbb R^{+}\cong\mathbb R^{*}/\mathbb Z_2$$
A: For $(1)$, all you need to do is check the map you defined is in fact an isomorphism (check it's a homomorphism, check its image, and check its kernel)
In response to $(2)$, $H$ is the group of elements {$ 4^n| n\epsilon \mathbb Z $}. The multiplicative group $(\mathbb Q^+,*)$ is the group of the positive rational numbers under multiplacation in the usual way. H is normal since every subgroup of an abelian group is normal. You can represent each of the cosets with $m/n$ where $4$ does not divide $n$ or $m$. This representation is obviously unique and any two cosets have different representations of this form. Some elements in $G/H$ are of finite order some are not. This should be enough to do the problem.
A: For the first one it is easier if you do the following. First of all $\mathbb{Z}_2$ is isomorphic to $\{-1,1\}$ where the operation is multiplication. Call this group $G$. Then $\mathbb{R}^+\times\mathbb{Z}_2\cong \mathbb{R}^+\times G$. Then define $F(r,g)=g\cdot r$ (the usual multiplication). Then check that is is 1-1, onto, and a morphism: $F((r,g)(r',g'))=F(rr',gg')=gg'\cdot rr'=(gr)(g'r')=F(r,g)F(r',g')$. You can check the rest. For the other part Don Antonio just answered it. 
