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clooneyisagenius

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Registered: 2007-03-25

Posts: 56

Abstract Algebra Proof Help - Group Theory

Let G be the set of all real-valued functions on the interval [0,1]. Define f+g for f,g in G by (f+g)(x)=f(x)+g(x).

1. Prove that G is a group
2. Define phi : G->R by phi(f)=phi(1/4) and prove that phi is a homomorphism
3. Let H={f in G such that f(1/4)=0}. Prove H is a subgroup of G.
4. What is G/H isomorphic to?

All in all - just stuck. Any ideas?

My work so far:
G is a group means that it is assosciative, closed, has an inverse and identity. (It's associative because addition is)

JaneFairfax

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Registered: 2007-02-23

Posts: 6,868

Re: Abstract Algebra Proof Help - Group Theory

What do you mean by phi(1/4)?

clooneyisagenius

Member

Registered: 2007-03-25

Posts: 56

Re: Abstract Algebra Proof Help - Group Theory

phi is the permutation.  but on #2 i meant phi(f) = f(1/4)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Abstract Algebra Proof Help - Group Theory

Well, then #2 is straightforward, isn’t it?

#3 is also straightforward.

For #4, show that H is the kernel of the homomorphism φ, then use the isomorphism theorem on the factor group G/H.

Last edited by JaneFairfax (2007-12-17 04:01:23)

JaneFairfax

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Registered: 2007-02-23

Posts: 6,868

Re: Abstract Algebra Proof Help - Group Theory

To show that H is a group:

Closure: Let f,g ∈ H. Show that f+g ∈ H, i.e. (f+g)(1⁄4) = 0. Well, f,g ∈ H means f(1⁄4) = 0 and g(1⁄4) = 0. So all you need to do is just add them.

Associativity: You can just say that addition of functions in general is associative.

Identity: Show that the identity in G is in H.

Inverse: Show that if f ∈ H, then −f ∈ H.

Last edited by JaneFairfax (2007-12-17 05:11:24)

clooneyisagenius

Member

Registered: 2007-03-25

Posts: 56

Re: Abstract Algebra Proof Help - Group Theory

Thanks. I also got number one. Thanks for you're help on 3, it pushed me to finishing one. Hopefully doing this problem on the practice exam I'll be able to do a similar one on the final. You've been a huge help.