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

by jessicaw
Tags: quesyion
 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 $a\in G$, there exists a left inverse a' in G such that a'a=e." is enough.
 P: 313 Provided you have shown the multiplication is associative then the inverse has to be unique.
P: 56
 Quote by Simon_Tyler 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)?

Emeritus
PF Gold
P: 8,992

## "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-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.
 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?
P: 56
 Quote by 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?
I use (2) as my argument for (1). I know that (2) is valid so (2) implys (1)?
Emeritus