How to prove that a set is a group? related difficult/challenge quesyion.

[jessicaw]

"How to prove that a set is a group?" related difficult/challenge quesyion.

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 $a\in G$, there exists a left inverse a' in G such that a'a=e." is enough.

 

[Simon_Tyler]

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

 

[jessicaw]

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

so in proving a set is a group it is redundant to show the uniqueness of inverse(just show existence is ok)?

 

[Fredrik]

It's only unnecessary if you have already proved that the axioms (xy)z=x(yz), ex=x and x^-1x=e define a group. (The standard axioms are (xy)z=x(yz), ex=xe=x and x^-1x=xx^-1x=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 e-inverse" 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 e-inverse to x, then x is a left e-inverse 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 left-e inverse is a right e-inverse. (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.

 

[Landau]

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?

 

[jessicaw]

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?

I use (2) as my argument for (1). I know that (2) is valid so (2) implys (1)?

 

[Fredrik]

I use (2) as my argument for (1). I know that (2) is valid so (2) implys (1)?

Suppose that y and y' are left e-inverses of x, and z is a right e-inverse of x. Then y'x=yx, and if you multiply by z from the right, you get y'=y. Is that what you wanted to know? I'm not sure I understand what you're asking.

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.