Group (Associativity of Binary Operators)

In summary, the key element of a group is that its elements must be associative. This means that any three elements from the group should follow the rule a*(b*C) = (a*b)*c. However, this only holds true if c commutes with ab, which is not always the case. Therefore, the statement is not always true and a counterexample can be found in non-abelian groups.
  • #1
jeff1evesque
312
0
Statement:
One of the key elements in being in what people call group is that elements must be associative.

So this means if we take any three elements from what we propose to be a group, they should be associative,
a*(b*C) = (a*b)*c

Question:
Suppose we do have a group with elements a, b, c.
Would the following be true,
a*(b*C) = (a*b)*c = (c*a)*b
 
Physics news on Phys.org
  • #2
It would be true if and only if c commutes with ab. In that case, you only need associativity in a(bc) = (ab)c = c(ab) = (ca)b.
However, in general that is not true. Probably your favorite non-abelian group provides an easy counterexample.
 

1. What is the definition of associativity in regards to binary operators?

Associativity is a property of binary operators that determines the order in which operations are performed on a sequence of operands. In an associative operation, the grouping of operands does not affect the final result.

2. Why is associativity important in mathematical operations?

Associativity allows us to simplify complex mathematical expressions by grouping operands in a different order without changing the result. It also ensures that the result of an operation is independent of the order in which the operations are performed.

3. What is the difference between left-associative and right-associative operators?

Left-associative operators evaluate operations from left to right, while right-associative operators evaluate operations from right to left. This determines the order in which the operations are performed and can affect the final result.

4. Can non-associative operators exist?

Yes, non-associative operators do exist. In these cases, the order in which the operations are performed does affect the final result. Examples of non-associative operators include the subtraction and division operators.

5. How does associativity apply to real-life scenarios?

In real-life scenarios, associativity can be seen in various operations, such as addition, multiplication, and logical operations. For example, in a math problem like (2+3)+4, associativity allows us to group (2+3) first, and then add 4 to get the same result as grouping (3+4) first and then adding 2.

Similar threads

I Group like operations that are not associative
    • Linear and Abstract Algebra
Replies
3
Views
1K
I Does associativity imply bijectivity in group operations?
    • Linear and Abstract Algebra
2
Replies
38
Views
2K
I Number of Binary Operations on a Set with a Special Property
    • Linear and Abstract Algebra
Replies
2
Views
464
MHB Which of the group axioms are satisfied?
    • Linear and Abstract Algebra
Replies
9
Views
2K
MHB Determine if a group is abelian
    • Linear and Abstract Algebra
Replies
2
Views
1K
I Finding All Automorphisms of Group
    • Linear and Abstract Algebra
Replies
2
Views
1K
MHB Proving matrix group under addition for associative axiom
    • Linear and Abstract Algebra
Replies
1
Views
1K
MHB Proving $(G,*)$ is a Group: Hints and Tips for a Simple Group Theory Problem
    • Linear and Abstract Algebra
Replies
2
Views
1K
I Understanding Quotient Groups in Abstract Algebra
    • Linear and Abstract Algebra
Replies
8
Views
2K
I For groups, showing that a subset is closed under operation
    • Linear and Abstract Algebra
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top