Abelian group facts for kids

An abelian group is a special kind of group in group theory. It's called 'abelian' because its operations follow a rule called 'commutativity'. This means the order in which you combine things doesn't change the result. Imagine adding numbers: 2 + 3 is the same as 3 + 2. That's commutativity!

What is an Abelian Group?

An abelian group is made of two main parts:

• A set of items, let's call it A. A set is just a collection of different things.
• An operation, like adding or multiplying, which we'll call "•". This operation combines any two items from the set A to make a new item.

For a set A and an operation "•" to be an abelian group, they must follow five important rules. These rules are sometimes called the abelian group axioms.

The Five Rules

Here are the rules that A and "•" must follow:

Rule 1: Closure

When you combine any two items from the set A using the operation "•", the result must also be an item in A.

• Example: If A is the set of whole numbers and "•" is addition, then 3 + 5 = 8. Since 8 is also a whole number, this rule is met.

Rule 2: Associativity

When you combine three or more items, the way you group them doesn't change the final answer.

• Example: For any items a, b, and c in A, (a • b) • c must be the same as a • (b • c).
• Think of it like this: (2 + 3) + 4 = 5 + 4 = 9. And 2 + (3 + 4) = 2 + 7 = 9. The result is the same!

Rule 3: Identity Element

There must be a special item in the set A, let's call it e. When you combine e with any other item a using the operation "•", the item a doesn't change.

• Example: For addition, the identity element is 0, because 0 + 5 = 5 and 5 + 0 = 5.
• For multiplication, the identity element is 1, because 1 × 5 = 5 and 5 × 1 = 5.

Rule 4: Inverse Element

For every item a in the set A, there must be another item, let's call it b, also in A. When you combine a and b using the operation "•", the result is the identity element e.

• Example: For addition, the inverse of 5 is -5, because 5 + (-5) = 0 (the identity element).
• For multiplication, the inverse of 5 is 1/5, because 5 × (1/5) = 1 (the identity element).

Rule 5: Commutativity

This is the special rule that makes an abelian group different from other groups. The order in which you combine two items doesn't matter.

• Example: For any items a and b in A, a • b must be the same as b • a.
• Like we said before: 2 + 3 = 5 and 3 + 2 = 5. The order doesn't change the sum.

If a group follows all these five rules, it's called an "abelian group." If a group follows the first four rules but not the fifth (commutativity), it's called a "non-abelian group" or "non-commutative group."

See also

In Spanish: Grupo abeliano para niños