Group¶

Definition¶

A group is a set $A$ together with a binary operation $\star$ such that:

1. $\star$ is associative;
2. $\star$ has an identity element;
3. every element has an inverse element.

Interpretation¶

Associativity allows parentheses to move in $a\star b\star c$ without changing the result. This makes repeated group operations unambiguous; it does not allow the factors to be reordered.

Groups capture the algebra of reversible operations.

The table shows addition modulo $4$, which is associative and has identity $0$. Each row contains a green $0$, exhibiting an additive inverse; for example, $1+3=0$ and $2+2=0$$0$. Each row and column is a permutation of $\{0,1,2,3\}$. The symmetry of the table also shows that this group is Abelian.

Key Results¶

• Every group is a monoid.
• If the operation is also commutative, the group is Abelian.

Examples¶

• $(\mathbb{Z},+)$ is a group. $(\mathbb{N},+)$ is not a group because positive natural numbers do not have additive inverses in $\mathbb{N}$.

Prerequisites¶

• Associativity
• Identity Element
• Inverse Element

Related Concepts¶

• Binary Operation
• Algebraic Structure
• Commutativity
• Monoid
• Ring
• Semigroup

Sources¶

• Math II: Linear Algebra · Week 1

Symbols: integers, natural numbers

Last updated: 20 October 2026 03:02