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How to prove that a set is a group? related difficult/challenge quesyion.by jessicaw
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#1
Oct1910, 11:41 PM

P: 56

1.In proving that a set is a group from definition, we have to show there is an inverse element for each element, do we have to show uniqueness of it?
I believe this is unnecessary but people do this all the time. My argument: "For each [itex]a\in G[/itex], there exists a left inverse a' in G such that a'a=e." is enough. 


#2
Oct2010, 01:30 AM

P: 313

Provided you have shown the multiplication is associative then the inverse has to be unique.



#3
Oct2010, 07:28 AM

P: 56




#4
Oct2010, 11:32 AM

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How to prove that a set is a group? related difficult/challenge quesyion.
It's only unnecessary if you have already proved that the axioms (xy)z=x(yz), ex=x and x^{1}x=e define a group. (The standard axioms are (xy)z=x(yz), ex=xe=x and x^{1}x=xx^{1}x=e). This is a bit tricky. Define a "left identity" to be an element e such that ex=x for all x, and a "left einverse" of x to be an element y such that yx=e. You can prove the theorem by proving all of the following, in this order.
1. There's at most one right identity. 2. If y is a left einverse to x, then x is a left einverse of y. 3. e is a right identity (which by #1 must be unique). 4. There's at most one left identity. (This means that e is an identity and is unique). 5. Every lefte inverse is a right einverse. (Hint: Use 2). 6. Every x has at most one left inverse. (This means that every element has a unique inverse). If you've done this once, or if this is a theorem in your book, then all you have to do to verify that the structure you're considering is a group is to verify that the alternative axioms stated above are satisfied. 


#5
Oct2010, 03:14 PM

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P: 905

It seems two different questions are asked:
(1) Is it enough to show that each element has an inverse, instead of in addition showing this inverse is unique? (2) Is it enought to show that each element has a LEFT inverse, instead of showing that each element has both a right and a left inverse? Or is it (3) Is it enought to show that each element has a LEFT inverse, instead of showing that each element has both a right and a left inverse and that these are unique? 


#6
Oct2010, 11:01 PM

P: 56




#7
Oct2110, 06:01 AM

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P: 9,416

By the way, in both #1 and #6, you're referring to a mere statement as an "argument". It's not an argument if it doesn't include some evidence for the claim. 


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