Thread: Show that R^x/<-1> is isomorphic to the group of positive real numbers under multipli

Show that R^x/<-1> is isomorphic to the group of positive real numbers under multiplication.



I know I need to show we have a homomorphism, and is one - to one and onto in order to be isomorphic. I know all that, but I don't know what function mapping to use.

Originally Posted by kathrynmath

Show that R^x/<-1> is isomorphic to the group of positive real numbers under multiplication.



I know I need to show we have a homomorphism, and is one - to one and onto in order to be isomorphic. I know all that, but I don't know what function mapping to use.

An idea: check the function ...

Tonio

I'm curious as to how you chose that function?

Originally Posted by kathrynmath

I'm curious as to how you chose that function?

Well, you need a multiplicative function, i.e. and I don't know that many, so....

Tonio

So you can just choose any multiplicative function?
so assume f(x1)=f(x2)
|x1|=|x2|
x1=x2
1-1
onto because a maps to |a|
Now f(x)f(y)=|x||y|=|xy|=f(xy) homomorphism
Thus isomorphic.

Originally Posted by kathrynmath

So you can just choose any multiplicative function?


No, not any. Say, doesn't work. Do you know any other multiplicative functions for our problem?

Tonio


so assume f(x1)=f(x2)
|x1|=|x2|
x1=x2
1-1
onto because a maps to |a|
Now f(x)f(y)=|x||y|=|xy|=f(xy) homomorphism
Thus isomorphic.

.

Originally Posted by kathrynmath

So you can just choose any multiplicative function?
so assume f(x1)=f(x2)
|x1|=|x2|
x1=x2
1-1
onto because a maps to |a|
Now f(x)f(y)=|x||y|=|xy|=f(xy) homomorphism
Thus isomorphic.

No. My function is NOT an isomorphism since it is not 1-1, but you can apply the first isomorphism theorem for groups...

Tonio

If G is a group, N is a normal subgroup of G, and H is any subgroup of G. Then HN is a subgroup of G, H intersect N is a normal subgroup of H and HN/N isomorphic to H/(H intersectN) by the first isomorphism theorem.

I'm not quite sure how to use that.

Originally Posted by kathrynmath

If G is a group, N is a normal subgroup of G, and H is any subgroup of G. Then HN is a subgroup of G, H intersect N is a normal subgroup of H and HN/N isomorphic to H/(H intersectN) by the first isomorphism theorem.

I'm not quite sure how to use that.

That is not the standard first isomorphism theorem for groups, which you can read in one of the following sites:

Isomorphism theorem - Wikipedia, the free encyclopedia

http://www.tjsullivan.org.uk/math/morphthm.pdf (pages 5-6)

http://marauder.millersville.edu/~bi...o/firstiso.pdf

If you look at the above carefully then you'll understand why the absolute value function solves your problem.

Tonio