Groups Definition and Examples: Explained with Examples

A group is a set G with a binary operation satisfying four axioms: closure (a ∗ b ∈ G), associativity ((a ∗ b) ∗ c = a ∗ (b ∗ c)), an identity element e with a ∗ e = e ∗ a = a, and inverses (every a has b with a ∗ b = b ∗ a = e). The integers under addition, (ℤ, +), are the canonical example: 0 is the identity and −a is the inverse of a.

Groups are the mathematics of symmetry. Rotations of a square, permutations of objects, invertible matrices, and the number systems themselves are all groups. Whenever a collection of reversible transformations can be combined, a group is hiding underneath — which is why groups dominate algebra, geometry, and physics.

Key Concepts

• The four axioms: closure, associativity, identity, inverses — in that logical order.
• Abelian group: the operation also commutes (a ∗ b = b ∗ a); (ℤ, +) is abelian.
• Non-abelian group: order of combination matters; symmetries of a square form the non-abelian group D₄ of order 8.
• Order of a group: its number of elements; (ℤ, +) has infinite order, D₄ has order 8.
• Core examples: (ℤ, +), (ℝ, +), (ℝ − {0}, ×), the symmetric group Sₙ of permutations, cyclic groups Cₙ, dihedral groups Dₙ.
• Consequences: the identity is unique, each inverse is unique, (ab)⁻¹ = b⁻¹a⁻¹, and cancellation holds (ab = ac ⟹ b = c).

Formulas

• Axioms: (1) a ∗ b ∈ G; (2) (a ∗ b) ∗ c = a ∗ (b ∗ c); (3) ∃e ∀a: a ∗ e = e ∗ a = a; (4) ∀a ∃a⁻¹: a ∗ a⁻¹ = a⁻¹ ∗ a = e.
• Inverse of product: (ab)⁻¹ = b⁻¹a⁻¹ — note the order reversal, like taking off shoes then socks.
• Cancellation: left and right cancellation follow from multiplying by inverses.
• One-sided suffices: with associativity, a right identity and right inverses already force a group.

Worked Example

Show that the symmetries of an equilateral triangle form a group of order 6 (the dihedral group D₃).

1. List the symmetries: identity e; rotations r (120°) and r² (240°); three reflections s, rs, r²s across the three axes.
2. Closure: combining symmetries is again a symmetry — e.g. r followed by s is the reflection rs.
3. Associativity: inherited from function composition.
4. Identity: e (do nothing) fixes every symmetry.
5. Inverses: r⁻¹ = r² (120° + 240° = 360° = e ✓); each reflection is its own inverse (s² = e).
6. Non-abelian check: rotating then reflecting (rs) differs from reflecting then rotating (sr = r²s ≠ rs) — order matters, so D₃ is non-abelian.

Common Misconceptions

• "(ℤ, ×) is a group." No — only 1 and −1 have multiplicative inverses in ℤ; 2 has no integer reciprocal. (ℚ − {0}, ×) is a group, though.
• "Every group is abelian." D₃, D₄, S₃, and matrix groups are non-abelian; commutativity is a bonus, not a requirement.
• "Checking one-sided identity and inverses is not enough." It is enough, given associativity — a standard theorem students often distrust.

Real-World Applications

• Crystallography: the 230 space groups classify all possible crystal symmetries.
• Cryptography: elliptic-curve groups underlie modern public-key encryption.
• Physics: rotation groups describe angular momentum; gauge groups (U(1), SU(2), SU(3)) structure the Standard Model.
• Puzzles: the Rubik's cube group has about 43 quintillion elements — group theory analyses it.

A group is closure, associativity, identity, and inverses — four axioms capturing reversible symmetry. The triangle symmetries show every axiom in action plus non-commutativity, and the consequences (unique inverses, (ab)⁻¹ = b⁻¹a⁻¹, cancellation) follow with a few lines of algebra. Nearly every structure ahead — subgroups, quotients, homomorphisms — is built on this definition.