What Are Simple Groups?

The demo draws a Cayley graph: one node for each element g of the group, and an edge from g to g·s for each chosen generator s. The layout pulls nodes apart until distances in space match distances in the graph, and the shape that appears belongs to the group: A₄ settles into a truncated tetrahedron, S₄ into a truncated octahedron, and A₅, with generators of orders 2 and 5, into a soccer ball.

Colors show conjugacy classes: x and y are conjugate when y = gxg⁻¹ for some g. A subgroup N is normal exactly when gNg⁻¹ = N for every g, so a normal subgroup is always a union of whole conjugacy classes, and it must contain the identity. Each class size divides the order of the group, and by Lagrange's theorem so must the size of N.

For A₅ the class sizes are 1, 12, 12, 15 and 20. Adding 1 to any selection of the others gives 13, 16, 21, 25, 28, 33, 36, 40, 45, 48 or 60, and of these only 60 divides 60. So A₅ has no normal subgroups except {e} and itself: it is simple. PSL(2, 7), with class sizes 1, 21, 42, 56, 24 and 24, falls to the same counting argument.

When N is normal, its cosets gN partition the group into blocks of equal size, and they multiply as a group G/N. Merging each coset into one node turns the Cayley graph of G into the Cayley graph of G/N, with the images of the same generators. That is the collapse the demo animates.

A simple group is a group G that has exactly two normal subgroups: the trivial group {e} and G itself. In other words, G has no proper, non-trivial normal subgroups. This definition may seem modest, but it plays a role in group theory analogous to the role of prime numbers in number theory or atoms in chemistry: simple groups are the indivisible building blocks from which all finite groups are assembled.

Recall that a subgroup N of G is normal (written N ⊴ G) if gNg⁻¹ = N for every g ∈ G. Normal subgroups are exactly the kernels of group homomorphisms, so they are the subgroups we can "divide out" to form a quotient group G/N. A simple group is one where this factoring process cannot proceed further — there is no non-trivial way to decompose it into smaller pieces via quotients.

The analogy to prime numbers runs deep. Just as every positive integer greater than 1 can be uniquely factored into primes, every finite group can be broken down into simple groups through a process called a composition series. The simple groups that appear in this decomposition are unique up to order and isomorphism — this is the content of the celebrated Jordan-Holder theorem. The classification of finite simple groups therefore tells us the complete list of "primes" from which all finite groups are built.

The simplest examples of simple groups are the cyclic groups of prime order, Z/pZ. If p is prime, then Z/pZ has no subgroups other than {0} and itself (by Lagrange's theorem, any subgroup's order must divide p). Since every subgroup of an abelian group is automatically normal, Z/pZ is simple. In fact, these are the only abelian simple groups — every abelian simple group is isomorphic to Z/pZ for some prime p.

The symmetric group S₃ (the group of all permutations of three elements, with 6 elements) is not simple. It contains the alternating group A₃ ≅ Z/3Z as a normal subgroup of index 2. Similarly, Z/6Z is not simple because it contains the normal subgroups Z/2Z and Z/3Z. In general, any group of composite order that is abelian fails to be simple.

The first non-abelian simple group is the alternating group A₅, which has 60 elements. It is the group of even permutations of five objects, and proving its simplicity — that it has no normal subgroups of order 2, 3, 4, 5, 6, 10, 12, 15, 20, or 30 — is a classic exercise in algebra. This group also appears as the rotation symmetry group of the regular icosahedron, linking simplicity to geometric beauty.

Key insight: Abelian groups are simple only when they have prime order, since every subgroup of an abelian group is normal. Non-abelian simple groups are far rarer and more exotic — the smallest is A₅ with 60 elements. The gap between the smallest prime (2) and the smallest non-abelian simple group (60) hints at how special these objects are.

The Jordan-Holder theorem states that every finite group G admits a composition series — a chain of subgroups G = G₀ ⊵ G₁ ⊵ G₂ ⊵ ··· ⊵ Gₙ = {e} where each quotient Gᵢ/Gᵢ₊₁ is a simple group. Furthermore, the list of these simple quotients (called composition factors) is uniquely determined by G, up to reordering and isomorphism. This is the precise sense in which simple groups are the "primes" of group theory.

This theorem transforms the study of finite groups into two complementary programs. First, classify all finite simple groups — determine the complete list of "primes." Second, understand how simple groups can be assembled into larger groups — the extension problem. The first program was completed (in principle) with the classification theorem. The second remains one of the great open challenges of algebra: the number of groups of order n grows wildly and unpredictably as n increases.

The classification theorem tells us that every finite simple group is isomorphic to one of the following: (1) a cyclic group Z/pZ of prime order, (2) an alternating group Aₙ for n ≥ 5, (3) a group of Lie type (one of 16 infinite families of matrix-like groups over finite fields), or (4) one of exactly 26 sporadic groups that fit into none of the other families. This four-part answer is one of the most remarkable theorems in all of mathematics.