Left coset of a subgroup of Complex numbers.


by Silversonic
Tags: complex, coset, numbers, subgroup
Silversonic
Silversonic is offline
#1
Nov25-11, 08:03 PM
P: 126
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.
Phys.Org News Partner Science news on Phys.org
Cougars' diverse diet helped them survive the Pleistocene mass extinction
Cyber risks can cause disruption on scale of 2008 crisis, study says
Mantis shrimp stronger than airplanes
HallsofIvy
HallsofIvy is offline
#2
Nov26-11, 07:25 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,896
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
Silversonic is offline
#3
Nov26-11, 11:18 AM
P: 126
Quote Quote by HallsofIvy View Post
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+.
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
I like Serena is offline
#4
Nov26-11, 12:05 PM
HW Helper
I like Serena's Avatar
P: 6,189

Left coset of a subgroup of Complex numbers.


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


Register to reply

Related Discussions
Complex numbers representing Real numbers General Math 3
Symmetry breaking: what is the subgroup left? High Energy, Nuclear, Particle Physics 15
Symmetry breaking: what is the subgroup left? Linear & Abstract Algebra 0
Complex numbers - are they the 'ultimate', or are there any "complex complex" numbers Calculus 7
Elements in only 1 left/right coset Linear & Abstract Algebra 2