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

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In summary: For example, in #1, you say "Provided you have shown the multiplication is associative then the inverse has to be unique." But you don't provide any evidence for that. In #6, you say "Every x has at most one left inverse." But you don't provide any evidence for that.
  • #1
jessicaw
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"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 [itex]a\in G[/itex], there exists a left inverse a' in G such that a'a=e." is enough.
 
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  • #2


Provided you have shown the multiplication is associative then the inverse has to be unique.
 
  • #3


Simon_Tyler said:
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)?
 
  • #4


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.
 
  • #5


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


Landau said:
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)?
 
  • #7


jessicaw said:
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.
 

1. How do you define a group?

A group is a mathematical structure consisting of a set of elements and a binary operation that combines any two elements to form a third element, satisfying four axioms: closure, associativity, identity, and inverse.

2. What is the process of proving a set is a group?

The process of proving a set is a group involves verifying that the set satisfies the four group axioms: closure, associativity, identity, and inverse. This can be done by showing that the binary operation is well-defined, associative, has an identity element, and each element has an inverse.

3. What is closure and why is it important in proving a set is a group?

Closure is the property that states that the result of the binary operation on any two elements in the set must also be an element in the set. It is important in proving a set is a group because it ensures that the operation is well-defined and allows for the group to be closed under the operation.

4. Are there any exceptions to the group axioms?

Yes, there are exceptions to the group axioms. For example, a group can be non-commutative (not satisfy the commutative property), or non-associative. However, these exceptions are not considered to be true groups and are instead classified as other mathematical structures.

5. Can a set be a group if it does not have an identity element?

No, a set cannot be a group if it does not have an identity element. The identity element is a crucial part of the group structure and without it, the other axioms cannot be satisfied. A set without an identity element may still exhibit some group-like properties, but it would not be considered a group.

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