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Group (Mathematics) — Definition, Formula & Examples

A group is a set paired with an operation that combines any two elements to produce a third element, subject to four rules: closure, associativity, the existence of an identity element, and the existence of inverses for every element.

A group is an ordered pair (G,∗)(G, \ast) where GG is a non-empty set and ∗ ⁣:G×G→G\ast\colon G \times G \to G is a binary operation satisfying: (1) associativity — (a∗b)∗c=a∗(b∗c)(a \ast b) \ast c = a \ast (b \ast c) for all a,b,c∈Ga, b, c \in G; (2) identity — there exists e∈Ge \in G such that e∗a=a∗e=ae \ast a = a \ast e = a for all a∈Ga \in G; (3) inverses — for each a∈Ga \in G there exists a−1∈Ga^{-1} \in G such that a∗a−1=a−1∗a=ea \ast a^{-1} = a^{-1} \ast a = e. Closure is implied by ∗\ast being a binary operation on GG.

How It Works

To verify that a set with an operation forms a group, you check all four axioms one by one. First confirm the operation always produces an element still in the set (closure). Then confirm associativity holds — note this does not require commutativity. Next, identify the identity element, and finally show every element has an inverse under the operation. If the operation is also commutative, the group is called abelian.

Worked Example

Problem: Show that the integers modulo 4 under addition, (Z4,+)(\mathbb{Z}_4, +) where Z4={0,1,2,3}\mathbb{Z}_4 = \{0, 1, 2, 3\}, form a group.
Closure: Adding any two elements and taking the remainder mod 4 always yields an element in {0, 1, 2, 3}. For instance, 2 + 3 = 5, and 5 mod 4 = 1, which is in the set.
2+3≡1(mod4)2 + 3 \equiv 1 \pmod{4}
Associativity: Addition of integers is associative, and this property carries over to addition mod 4.
(a+b)+c≡a+(b+c)(mod4)(a + b) + c \equiv a + (b + c) \pmod{4}
Identity: The element 0 serves as the identity because adding 0 to any element leaves it unchanged.
a+0≡a(mod4)a + 0 \equiv a \pmod{4}
Inverses: Each element has an inverse: 0 is its own inverse, the inverse of 1 is 3, the inverse of 2 is 2, and the inverse of 3 is 1.
1+3≡0(mod4),2+2≡0(mod4)1 + 3 \equiv 0 \pmod{4}, \quad 2 + 2 \equiv 0 \pmod{4}
Answer: All four axioms are satisfied, so (Z4,+)(\mathbb{Z}_4, +) is a group. It is also abelian since addition is commutative.

Why It Matters

Groups are the backbone of abstract algebra and appear throughout mathematics, physics, and computer science. Symmetry groups describe molecular structures in chemistry and particle interactions in physics. Understanding groups is a prerequisite for studying rings, fields, and Galois theory in upper-level algebra courses.

Common Mistakes

Mistake: Assuming every group is commutative (abelian).
Correction: The group axioms do not require a∗b=b∗aa \ast b = b \ast a. Many important groups, such as the symmetric group S3S_3, are non-abelian. Always check commutativity separately.