Proving Abelian Group Structure in C^A

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In summary, the conversation is discussing two questions related to proving the group structure of a ring, specifically in the context of composition of functions and the properties of group homomorphisms. The first question asks for clarification on how to prove the four axioms of a ring using the given information, while the second question seeks guidance on using composition of functions and the identity function to prove the group structure. The conversation also briefly mentions the need to show closure and associativity in the proof.
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Homework Statement



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Homework Equations





The Attempt at a Solution



For the first question, since [f(a)][g(a)] is in C, can I just say that since C is a ring, it is an abelian group, then the four axioms are proven? Then just show closure? Probably not I'm guessing. Associativity of multiplying functions seems so fundamental to me that I really don't know what to do...

For the second question, shall I start by using composition of functions and then the properties of composition of a function and its inverse, and then go on about the identity function?
 

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I'd say question one is asking you to flesh out the group structure.

1. What's the identity element? (make sure it's a member of C^A).

2. For each element f, what's the inverse element? (make sure it's well defined
- namely, why can you find f^-1 for each f? )

Closure is trivial but should be mentioned, namely why is (fg) a well
defined member of C^A when f and g are?

Associativity (and Abelian-ness) of course follow from the properties of C,
but you should have a proof showing f(gh) =(fg)h.

For question 2. I don't know what A-hat is. What do you know
about group homomorphisms at this point?
 

FAQ: Proving Abelian Group Structure in C^A

What does it mean for a group to be abelian?

A group is considered abelian if its operation is commutative, meaning the order in which elements are combined does not affect the result. In other words, for any two elements a and b in the group, a*b = b*a.

How do you prove that a group is abelian?

To prove that a group is abelian, you must show that the group's operation is commutative. This can be done by demonstrating that for any two elements a and b in the group, a*b = b*a. If this condition holds true, the group is considered abelian.

Are all groups abelian?

No, not all groups are abelian. In fact, most groups are non-abelian. A group is only considered abelian if its operation is commutative, which is not a requirement for all groups.

Can a group be both abelian and non-abelian?

No, a group cannot be both abelian and non-abelian. A group is either abelian or non-abelian based on the commutativity of its operation. It cannot have both properties simultaneously.

What are some common examples of abelian groups?

Some common examples of abelian groups include the integers under addition, the set of real numbers under addition, and the group of 2x2 invertible matrices under matrix multiplication. Any cyclic group is also abelian.

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