Cosocle

The cosocle of a finite group G is the intersection of all its maximal normal subgroups. It is the lattice-dual of the socle: where the socle is the join of the minimal normal subgroups, the cosocle is the meet of the maximal ones. It is characteristic.

Definition

A maximal normal subgroup M is a proper normal subgroup with no normal subgroup strictly between M and G — equivalently, G/M is simple. The cosocle is their intersection:

⋂ { M : M a maximal normal subgroup of G }.

G / cosocle(G) is the largest quotient of G that is a subdirect product of simple groups — it embeds in the product of the simple quotients G/M.

Properties

• Characteristic, hence normal: an automorphism permutes the maximal normal subgroups and so fixes their intersection.
• Dual to the socle. For a simple group the only maximal normal subgroup is {e}, so cosocle(G) = {e} while soc(G) = G — the two swap roles. More generally cosocle(G) = {e} exactly when G is a subdirect product of simple groups (e.g. A₅ × A₅).
• Frattini for abelian groups. When every subgroup is normal — in particular for abelian groups and for p-groups — the maximal normal subgroups are exactly the maximal subgroups, so the cosocle equals the Frattini subgroup Φ(G).
• Contains the derived subgroup. Each G/M is simple; the abelian ones force G' ≤ M, so G' lies in the cosocle whenever the quotient G/cosocle(G) is abelian.

Examples

The bottom two rows show the duality plainly: for a group that is (a product of) simple groups the socle is everything and the cosocle is trivial.

See also

• Socle — the join of the minimal normal subgroups; the cosocle is its dual.
• Frattini subgroup — the intersection of the maximal subgroups; equals the cosocle for abelian and p-groups.
• Prime radical — the socle without the normality restriction.
• Simple group — the quotients G/M at the maximal normal subgroups.