Group theory · Definition

Binary operation

operationsgroups

Before you start you need to know

A binary operation combines two elements of a set and produces another element of the same set. It is the basis for studying groups, rings and other algebraic structures.

Definition

A binary operation on a set G is a function that takes any two elements of G and returns an element of G.

\[\ast:G\times G\to G\]

Also called: internal composition law

Closure is what makes an operation binary: for all \( a,b \in G \), the result \( a \ast b \) must lie in \( G \). Associativity and commutativity are not required.

On \( \mathbb{Z} \), addition is a binary operation because the sum of two integers is an integer, for example \( 3 + 5 = 8 \in \mathbb{Z} \).

A common mistake is forgetting that the result must belong to \( G \): subtraction on \( \mathbb{N} \) fails, since \( 2 - 5 = -3 \notin \mathbb{N} \).

Think about it

What does closure of an operation on G require?

That \( a \ast b \in G \) for all \( a,b \in G \).

Exercises 10

  1. A binary operation on G must be a function...
  2. Which is a binary operation on ℤ?
  3. To be a binary operation, the result must...
  4. The usual subtraction in ℤ is a binary operation because...
  5. What are chosen as inputs of a binary operation?
  6. The closure of an operation on G requires that...
  7. Associativity is written as...
  8. Which rule is a binary operation on ℤ?
  9. Which of these properties is NOT required for an operation to be binary on G?
  10. Which statement is false in general?

What comes next

References

  • I. N. Herstein, Topics in Algebra, 2.ª edición, John Wiley & Sons, 1975.