Simple groups

A nontrivial group is simple when its only normal subgroups are the trivial subgroup {e} and the whole group. The finite simple groups — cyclic of prime order, the alternating groups Aₙ (n ≥ 5), the groups of Lie type, and 26 sporadic groups — are completely classified.

Definition

A group G with |G| > 1 is simple if its only normal subgroups are {e} and G. Equivalent statements (all for |G| > 1):

• G has exactly two normal subgroups.
• Every quotient of G is trivial or G itself.
• Every homomorphism out of G is either trivial or injective (its kernel, being normal, is {e} or G).
• G is not a nontrivial extension: it has no proper nontrivial normal subgroup to quotient by.

The two worlds

Abelian simple groups are exactly the cyclic groups Cₚ of prime order. In an abelian group every subgroup is normal, so simplicity forces the only subgroups to be {e} and G — which happens iff |G| is prime.

A non-abelian simple group G satisfies all of:

• Perfect. G = [G, G]: the commutator subgroup is normal and nontrivial (as G is non-abelian), hence equals G. Its abelianization G / [G, G] is therefore trivial, and G is not solvable.
• Even order — every group of odd order is solvable (Feit–Thompson).
• At least three distinct prime divisors. A group of order pᵃqᵇ is solvable (Burnside), so |G| cannot use only one or two primes. The smallest, A₅, already has |A₅| = 2²·3·5.

Classification of finite simple groups

Every finite simple group is isomorphic to exactly one of:

This is the Classification of Finite Simple Groups (CFSG), one of the largest theorems in mathematics. The Lie-type groups are finite matrix groups over finite fields; the 26 sporadic groups fit none of the infinite families.

Why they matter — building blocks

Every finite group G has a composition series

{e} = G0 ◁ G1 ◁ ⋯ ◁ Gk = G,

a chain of normal steps whose successive quotients Gᵢ₊₁ / Gᵢ are all simple. By the Jordan–Hölder theorem these composition factors are unique up to reordering and isomorphism — independent of the chosen series (see chief series). Simple groups are therefore the irreducible "building blocks" from which every finite group is assembled by repeated extension.

This links the two extremes: a finite group is solvable iff every composition factor is cyclic of prime order — i.e. iff no non-abelian simple group appears. Nonabelian simple groups are exactly the obstruction to solvability.

Examples

• Cₚ — C₂, C₃, C₅, C₇, … the abelian simple groups, one for each prime.
• A₅ — order 60, the smallest non-abelian simple group. It is alternating, and also A₅ ≅ PSL(2,𝔽4) ≅ PSL(2,𝔽5).
• PSL(2,𝔽7) — order 168, the second-smallest non-abelian simple group, with PSL(2,𝔽7) ≅ PSL(3,𝔽2) ≅ GL(3,𝔽2).
• A₆ — order 360, with A₆ ≅ PSL(2,𝔽9).

Non-examples

• Cₙ with n composite — e.g. C₆ has the proper nontrivial normal subgroups C₂ and C₃. Only prime-order cyclic groups are simple.
• S₅ — order 120, not simple: A₅ is a normal subgroup of index 2. (Its only simple composition factor is A₅.)
• Any nontrivial solvable group of order pᵃqᵇ — e.g. S₄ (order 24 = 2³·3): solvable, so not non-abelian simple, and not of prime order.

See also

• Alternating group — Aₙ is simple for n ≥ 5.
• Perfect group — every non-abelian simple group is perfect.
• Almost simple group — the groups S ≤ G ≤ Aut(S) sitting over a non-abelian simple S.
• Matrix groups — the groups of Lie type.
• Cyclic group — the abelian simple groups Cₚ.
• Solvable group — the opposite extreme: only prime-order composition factors.
• Burnside's theorem — order pᵃqᵇ ⇒ solvable.
• Wikipedia: Simple group
• Wikipedia: List of finite simple groups