Almost simple groups

A finite group G is almost simple when it is sandwiched between a non-abelian simple group and that group's automorphism group:

S ≤ G ≤ Aut(S),     S non-abelian simple,

where S is identified with its image Inn(S) ≤ Aut(S). Such a G is a simple group with some of its symmetries adjoined — S itself at one end, all of Aut(S) at the other.

Equivalent definition

G is almost simple iff its socle soc(G) is a non-abelian simple group.

One direction is the sandwich itself; the other is forced. If soc(G) = S is simple and non-abelian, it is the unique minimal normal subgroup, so every non-trivial normal subgroup contains it. The centralizer C_G(S) is normal and meets S in Z(S) = {1}, so it cannot contain S — hence C_G(S) = {1}, and conjugation

G → Aut(S),     g ↦ (s ↦ g s g⁻¹)

is injective. Its image contains Inn(S) ≅ S, giving S ≤ G ≤ Aut(S).

Two consequences worth naming:

• Z(G) = {1} — the center lies in C_G(soc(G)).
• G is directly indecomposable: a non-trivial A × B has minimal normal subgroups inside both factors, so its socle is never simple.

Which groups sit over a given S

The subgroups between S and Aut(S) correspond exactly to the subgroups of the outer automorphism group

Out(S) = Aut(S) / Inn(S),

by G ↦ G/S. Up to isomorphism the almost simple groups with socle S correspond to the conjugacy classes of subgroups of Out(S): an isomorphism between two of them must carry socle to socle (the socle is characteristic), so it is realized by conjugation inside Aut(S).

Out(S) is always solvable — the Schreier conjecture, known as a consequence of the classification of finite simple groups — and in practice tiny, so each simple group carries only a handful of almost simple groups above it.

Examples

• Every non-abelian simple group, taking G = S: the chain S ≤ S ≤ Aut(S) is legitimate, and soc(S) = S. So A₅ and PSL(2,𝔽7) are almost simple.
• S₅ — order 120. Aut(A₅) = S₅ and Out(A₅) = C₂, so A₅ and S₅ are the only two almost simple groups with socle A₅.
• PGL(2,𝔽7) — order 336, the full Aut(S) for S = PSL(2,𝔽7), whose Out is C₂.
• A₆ carries four. Here Out(A₆) = C₂ × C₂, whose three subgroups of order 2 are pairwise non-conjugate (the group is abelian), so besides A₆ and Aut(A₆) = PΓL(2,𝔽9) of order 1440 there are three distinct groups of order 720 with socle A₆: S₆, PGL(2,𝔽9), and the Mathieu group M₁₀. They are pairwise non-isomorphic — an order alone never names an almost simple group.

Non-examples

• A₅ × C₂ — order 120, the same order as S₅. Its socle is A₅ × C₂, not simple; equivalently its center is C₂ ≠ {1}. Matching the order of an almost simple group is not enough.
• A₅ × S₃ — order 360. The center is trivial and the derived subgroup chain ends at the simple A₅, yet the socle is A₅ × C₃: the S₃ factor centralizes A₅, so the conjugation map to Aut(A₅) is not injective.
• SL(2,𝔽5) — order 120 again. Its socle is the center {±I} ≅ C₂. It is quasisimple, a cover of A₅ rather than an extension of it.
• 2⁴:A₅ — the socle is the elementary abelian normal subgroup 2⁴. A non-trivial abelian normal subgroup always rules the group out.

See also

• Simple group — the socle S.
• Socle — the invariant that decides the property.
• Monolithic group — the weaker property of a single minimal normal subgroup, simple or not; almost simple is the non-abelian simple case.
• Automorphism group — the ceiling Aut(S), and Out(S), which indexes the groups over S.
• Quasisimple group — the other way to sit next to a simple group: a central cover below S rather than an extension above it.
• Center — trivial for every almost simple group.
• Wikipedia: Almost simple group