Register to reply 
How to prove that a set is a group? related difficult/challenge quesyion.by jessicaw
Tags: quesyion 
Share this thread: 
#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

Emeritus
Sci Advisor
PF Gold
P: 9,224

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

Sci Advisor
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

Emeritus
Sci Advisor
PF Gold
P: 9,224

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. 


Register to reply 
Related Discussions  
Why the thread difficult looking for medical physics jobs is closed?  Forum Feedback & Announcements  2  
Ever attempt a food challenge ? Just looking for a laugh at the moment?  Fun, Photos & Games  1  
Qs: How Group and Period affect semiconductors  Atomic, Solid State, Comp. Physics  6 