Left coset of a subgroup of Complex numbers.

Silversonic

1. The problem statement, all variables and given/known data

For H [itex]\leq[/itex] G as specified, determine the left cosets of H in G.

(ii) G = [itex]\mathbb{C}[/itex]* H = [itex]\mathbb{R}[/itex]*

(iii) G = [itex]\mathbb{C}[/itex]* H = [itex]\mathbb{R}[/itex][itex]_{+}[/itex]





3. The attempt at a solution

I have the answers, it's just a little inconsistency I don't understand.

For (ii) left cosets are

{r(cos∅ + isin∅); r [itex]\in[/itex] (0,∞)} ∅ [itex]\in[/itex] [0, 2[itex]\pi[/itex])

For (iii)

{r(cos∅ + isin∅); r [itex]\in[/itex] [itex]\mathbb{R}[/itex] \ {0} } ∅ [itex]\in[/itex] [0, [itex]\pi[/itex])


I'm told that the answers are different because the range of r and ∅ are different. It says in (ii) they are "half lines" coming out of the origin and in (iii) they are lines through the origin but excluding the origin itself.


What I don't get, though, is that surely the answer for (ii) should be the answer for (iii)? And vice versa? Basically in (ii) we have H is the set of all the real numbers, while G is the set of all the complex numbers. So when we multiply an element of H by an element of G (and the constant multiplying the euler's forumla is positive), surely r would then range over all the real numbers (excluding zero).

Yet in (iii) we have H is the set of all the positive real numbers, while G is still the set of all the complex numbers (and the constant multiplying the euler's forumla is positive). So when we multiply an element of H by an element of G, surely r would only range over the positive real numbers, as opposed to all the real numbers exluding zero like the answer says?

Does anyone understand my problem?

Thanks.

HallsofIvy

The question does not even make sense. R+ is NOT a subgroup of C*. Are you sure you have copied the problem correctly? R+ is a subgroup of C+.

Silversonic

Yeah, I've double checked and I've copied it correctly.

When saying R+, I assumed it was talking about the multiplication of positive real numbers (as opposed to addition of R+, which cannot be a group let alone a subgroup).

Why would R+ under multiplication not be a subgroup of C*? Surely every possible value of R+ on the positive real line is some form a complex number. All R+ is in C*, as well as gh (where g and h are elements of R+) are elements of R+, and lastly the inverse of an element g, is 1/g which is in R+).

I like Serena

Yes, you are right. R+ is a group under multiplication.
And I agree, solutions (ii) and (iii) should be swapped around.



Here's my interpretation of (ii).

Let's consider one specific element of C*, say z=a cis(phi).
Multiply it with R* to get the left coset.
This is a line excluding zero.

Now if we consider z=a cis(phi + pi) we get the same coset.
Indeed if pick any z in C* on the line, we get the same coset.
So the coset is uniquely defined by an angle between 0 and pi, but is independent of a.

So a specific coset is: {r cis(phi) | r in R*} 0 ≤ phi < pi