Socle
The socle soc(G) of a finite group G is the subgroup generated by all
of its minimal normal subgroups. It is the "bottom layer" of G:
a characteristic subgroup that is a
direct product of simple groups.
Definition
A minimal normal subgroup of G is a non-trivial normal subgroup that
properly contains no smaller non-trivial normal subgroup of G. The socle
is the join of them all:
soc(G) = ⟨ M : M a minimal normal subgroup of G ⟩.
This join is internal-direct: soc(G) = M₁ × M₂ × ⋯ × Mₖ for some subset of
the minimal normal subgroups. Each minimal normal subgroup is
characteristically simple — its only characteristic subgroups are {e} and
itself, forcing it to be a direct power Tr of a single
simple group T. So soc(G) is a direct product of simple
groups. (The trivial group is its own socle, an empty product.)
Properties
- Characteristic.
soc(G) char G: an automorphism permutes the minimal normal subgroups, hence fixes their join. In particularsoc(G) ⊴ G. - Non-trivial. Every non-trivial finite group has a minimal normal
subgroup, so
soc(G) ≠ {e}wheneverG ≠ {e}. - Simple groups. If
Gis simple — or more generally characteristically simple,G ≅ Tk— thenGis its own only minimal normal subgroup andsoc(G) = G. - Solvable groups. If
Gis solvable every minimal normal subgroup is an elementary abelianp-group (over various primesp), sosoc(G)is abelian — a direct product of elementary abelianp-groups — and sits inside the Fitting subgroup,soc(G) ≤ F(G). Conversely a non-abelian socle certifies thatGis not solvable. - Abelian groups. For finite abelian
G,soc(G)is the largest subgroup of square-free exponent — equivalently the elements of square-free order. For ap-group this is the bottom layerΩ1(G) = { g : gp = e }. - Primitive permutation groups. The socle governs the O'Nan–Scott theorem: a finite primitive permutation group has socle a direct power of a simple group, and its type organizes the classification.
Examples
G | minimal normal subgroup(s) | soc(G) |
|---|---|---|
C₄ | the order-2 subgroup | C₂ |
C₂ × C₄ | the order-2 subgroups | C₂ × C₂ |
C₆ | C₂ and C₃ | C₆ |
S₃ | A₃ = C₃ | C₃ |
D₄ | center Z = C₂ | C₂ |
S₄ | V₄ | C₂ × C₂ |
A₅ | A₅ | A₅ |
S₅ | A₅ | A₅ |
D₄ and Q₈ share the socle C₂ — each has a single minimal
normal subgroup, its center — so the socle sees only the bottom layer and does
not determine the group. A non-abelian socle, such as A₅ inside S₅ or
the direct power A₅ × A₅ (its own socle), is the hallmark of a non-solvable
group.
See also
- Chief series — chief factors are minimal normal subgroups of successive quotients.
- Fitting subgroup — largest nilpotent normal subgroup; hosts the socle of a solvable group.
- Simple group — the pieces the socle is a direct product of.
- Monolithic group — exactly the groups whose socle is itself a minimal normal subgroup.
- Almost simple group — exactly the groups whose socle is a single non-abelian simple group.
- Characteristic subgroup — the socle is one.
- Wikipedia: Socle