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envision October 3rd, 2009 06:39 PM

Subgroup/Normal Subgroup/Factor Group Questions
 
Let G be the positive reals under multiplication and let H be numbers 2^i where i is in Z.

(a) Show H is a subgroup of G.
(b) Show H is a Normal subgroup of G.
(c) Give a natural representation of the factor group G/H. By this I mean that each element should be uniquely describable as a-bar where a ranges over some natural set. (So, for example, you couldn’t have 3.1 and 12.4 as 3.1-bar = 12.4-bar.) Have the identity represented as 1-bar.
(d) Find all elements a-bar in G/H whose cube (in G/H) is the identity.
(This is a bit tricky. There are precisely three solutions!)

cknapp October 4th, 2009 03:24 AM

Re: Subgroup/Normal Subgroup/Factor Group Questions
 
Quote:

Originally Posted by envision
Let G be the positive reals under multiplication and let H be numbers 2^i where i is in Z.

(a) Show H is a subgroup of G.
(b) Show H is a Normal subgroup of G.

You should be able to do both of these by now if you are looking at Factor Groups. What are you having trouble with?

Quote:

(c) Give a natural representation of the factor group G/H. By this I mean that each element should be uniquely describable as a-bar where a ranges over some natural set. (So, for example, you couldn’t have 3.1 and 12.4 as 3.1-bar = 12.4-bar.) Have the identity represented as 1-bar.
We know that any element of the form 2^i is in H, so 2^i-bar = 1-bar. Now, choose some element r of R+. (2^i)*r = (i-bar)*r=r-bar. You want to choose a "nice" r. Can you take it from here? (hint: look at 2^0 and 2^1)
Quote:

(d) Find all elements a-bar in G/H whose cube (in G/H) is the identity.
(This is a bit tricky. There are precisely three solutions!)
You know one element straight off the bat. Choose a representative of 1-bar. There are 2 natural representatives. Look at the cube roots of these. If you chose your r-bars correctly, they will be the canonical representatives.


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