Cancellation Law in Group Theory: Proof and Validity

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In summary, the cancellation law will work if a*b=a*c. If a*b=c*a, it may not be true that b=c. Instead, a b a^-1 = c ( for example, consider the permutation compositions ( 1 2 ) ( 2 3 1 ) = ( 1 3 2 ) ( 1 2 ) , but ( 2 3 1 ) != ( 1 3 2 ) )It would be a useful exercise for you to find your mistake in your proof by applying your proof to wis' counterexample. Going wrong somewhere is always useful as long as you find out why.
  • #1
dijkarte
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Let a, b, and c be elements of group <G, *>, then
Can we apply cancellation law here:
a*b = c*a?

I could prove we can cancel a. However, some algebra
Texts do not state this which makes me
Uncertain about the validity of my proof.
 
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  • #2
dijkarte said:
Let a, b, and c be elements of group <G, *>, then
Can we apply cancellation law here:
a*b = c*a?

I could prove we can cancel a. However, some algebra
Texts do not state this which makes me
Uncertain about the validity of my proof.

The cancellation law will work if a*b = a*c. If a*b = c*a, it may not be true that b = c. Instead, a b a^-1 = c ( for example, consider the permutation compositions ( 1 2 ) ( 2 3 1 ) = ( 1 3 2 ) ( 1 2 ) , but ( 2 3 1 ) != ( 1 3 2 ) )
 
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  • #3
It would be a useful exercise for you to find your mistake in your proof by applying your proof to wis' counterexample. Going wrong somewhere is always useful as long as you find out why.

Note that the cancellation law you stat does hold for commutative groups! Maybe your proof uses commutativity somewhere?
 
  • #4
Here's my proof: Assuming a group <G, *>

Let a * b = c * a where a, b, c, all belong to G

Let a' denote inverse of a w.r.t the operation *

Let e denote identity for the operation *

a' * (a * b) = a' * (c * a)

Since an inverse exists for all elements of the group G, we can write a' * a = a * a' = e
So a' can be applied on both left and right.

Hence,

a' * (a * b) = (c * a) * a'

(a' * a) * b = c * (a * a') (applying associativity on both sides)

e * b = c * e ==> b = c

Is this correct?
 
  • #5
No, it's not correct because wis already gave you a counterexample. Apply wis his counterexample to your proof to see where it went wrong.
 
  • #6
dijkarte said:
Here's my proof: Assuming a group <G, *>

Let a * b = c * a where a, b, c, all belong to G

Let a' denote inverse of a w.r.t the operation *

Let e denote identity for the operation *

a' * (a * b) = a' * (c * a)

Since an inverse exists for all elements of the group G, we can write a' * a = a * a' = e
So a' can be applied on both left and right.

No, how do you decide you can apply a' both left and right? If a' commutes with every element of G, so does a; and since a was arbitrary, all groups are Abelian. Of course that's not true.
 
  • #7
BY definition of inverse for an operation * on a set, when we say an inverse exists, then we mean the inverse is applicable to both sides of the element. a' * a = a * a' = e.

By definition of a group (not necessarily commutative), an inverse exists such that
a' * a = a * a' = e, for all elements of the group.
 
  • #8
dijkarte said:
BY definition of inverse for an operation * on a set, when we say an inverse exists, then we mean the inverse is applicable to both sides of the element. a' * a = a * a' = e.

By definition of a group (not necessarily commutative), an inverse exists such that
a' * a = a * a' = e, for all elements of the group.

Yes, that's true.

But because a*a'=a'*a does NOT mean that a'*b=b*a' for all b!
 
  • #9
I see my mistake, thanks a lot for all. :)
 

FAQ: Cancellation Law in Group Theory: Proof and Validity

What is the cancellation law in group theory?

The cancellation law in group theory states that if two elements in a group have the same result when multiplied on both sides of an equation, then they are equal. In other words, if ab = ac, then b = c.

How is the cancellation law proven?

The cancellation law is typically proven using the properties of groups, such as associativity and the existence of an identity element. A common method is to assume that b and c are not equal, and then show that this leads to a contradiction.

Is the cancellation law valid in all groups?

Yes, the cancellation law is valid in all groups. This is because the definition of a group includes the existence of an inverse element for every element, which allows for the cancellation of elements on both sides of an equation.

Can the cancellation law be extended to other algebraic structures?

No, the cancellation law only applies to groups and cannot be extended to other algebraic structures. This is because other structures may not have the necessary properties for the cancellation law to hold.

How is the cancellation law used in practice?

The cancellation law is frequently used in algebraic equations to simplify expressions and solve for variables. It is also used in abstract algebra to prove theorems and properties of groups.

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