Chief series

A chief series is a maximal chain of normal subgroups of G. The chief factors are the consecutive quotients, and they are the coarsest canonical invariant of a finite group — two groups with different chief factor multisets are non-isomorphic.

Definitions

A normal series for G is a chain

{e} = G₀ ⊴ G₁ ⊴ G₂ ⊴ ⋯ ⊴ Gₖ = G

with each Gᵢ ⊴ G (every term normal in the full group, not merely in its successor). The quotients Gᵢ / Gᵢ₋₁ are the factors of the series.

A chief series is a normal series that cannot be refined: no G-normal subgroup lies strictly between Gᵢ₋₁ and Gᵢ. Equivalently, each chief factor Gᵢ / Gᵢ₋₁ is a minimal normal subgroup of G / Gᵢ₋₁.

By contrast, a composition series requires Gᵢ ⊴ Gᵢ₊₁ (not necessarily in G) and has simple factors. Composition series typically have more terms than chief series; chief series are often more useful for structural reasoning.

Properties of chief factors

Each chief factor Gᵢ / Gᵢ₋₁ is a minimal normal subgroup of a quotient of G. The structure of such a subgroup is sharp:

Theorem. A minimal normal subgroup of a finite group is a direct power of a single simple group:

M  ≅  T × T × ⋯ × T     (r copies, for some finite simple group T).

If T is abelian (i.e. T ≅ C_p for some prime p), the chief factor is an elementary abelian p-group.
If T is non-abelian simple, the chief factor is a non-abelian characteristically simple group.

Chief factors of solvable groups

For a solvable group G, every chief factor is an elementary abelian p-group (various primes p). This is equivalent to solvability: G is solvable iff its chief factors are all elementary abelian.

The multiset of primes that appear (with multiplicities = p-rank of each factor) determines |G| exactly.

Jordan-Hölder for chief series

Any two chief series of G have the same length and the same multiset of chief factors up to isomorphism and reordering. This is the analogue of the Jordan-Hölder theorem for composition series.

Examples

C₁₂. One chief series: {e} ⊴ C₂ ⊴ C₄ ⊴ C₁₂, factors C₂, C₂, C₃. Another: {e} ⊴ C₃ ⊴ C₆ ⊴ C₁₂, factors C₃, C₂, C₂. By Jordan-Hölder both have the same multiset of factors {C₂, C₂, C₃}.
S₄. Chief series {e} ⊴ V₄ ⊴ A₄ ⊴ S₄, factors V₄ = C₂ × C₂, C₃, C₂. V₄ appears as a single chief factor even though |V₄| = 4: the three order-2 subgroups of V₄ are permuted by S₄, so none is normal in S₄, so V₄ cannot be refined further.
A₅. {e} ⊴ A₅ is already a chief series — A₅ is simple. One chief factor: A₅.

Use in identification

Chief factors are a workhorse invariant when identifying a group:

Solvable vs non-solvable. Non-abelian chief factor ⟹ non-solvable (and flags a copy of a non-abelian simple group inside G).
Invariants. The sequence of chief-factor isomorphism types is a strong distinguishing invariant.
Structure inside the Fitting subgroup. Chief factors of solvable G lie inside F(G); their primes are the primes dividing |F(G)|.

Computation

Polycyclic group. Compute a normal-subgroup refinement of G into chief factors by splitting each layer into G-irreducible 𝔽ₚ-submodules. Each split is a linear-algebra problem over 𝔽ₚ.
Arbitrary finite group. Compute normal subgroups incrementally; refine any reducible chief factor by finding a proper G-normal subgroup.