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Cosocle

In group theory, a cosocle of a group G, denoted by Cosoc(G), is the intersection of all maximal normal subgroups of G.

Version
v1 · 2026-09-28 · History
Domain-specific #
8750
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomain
Group Theory → Mathematics

Core Idea

Cosocle is treated here as the recurring mathematics, logic, and statistics identity summarized by this source-grounded definition: In group theory, a cosocle of a group G, denoted by Cosoc(G), is the intersection of all maximal normal subgroups of G. In mathematics, the term cosocle (socle meaning pedestal in French) has several related meanings. In group theory, a cosocle of a group G, denoted by Cosoc(G), is the intersection of all maximal normal subgroups of G. If G is a quasisimple group, then Cosoc(G) = Z(G).

How would you explain it like I'm…

 

No faithful explanation at this level. Two of three generators judge that a child-level 'what all the biggest teams share' picture drops the normality condition and reads 'maximal' as 'biggest by size', describing a plain set overlap (or the Frattini-type intersection of all maximal subgroups) rather than the intersection of maximal NORMAL subgroups.

Shared Core of Special Teams

In math, a group is a collection of actions you can combine, like the ways to turn a square so it still looks the same. Some groups contain special smaller groups called normal subgroups. A maximal one is as large as possible without being the whole group. The cosocle of a group is what you get by keeping only the actions that sit inside every one of those maximal normal subgroups. It is the part they all share.

Intersection of Maximal Normal Subgroups

A group G is a set with a way of combining elements that follows certain rules. A normal subgroup is a subgroup that stays the same when you conjugate it by any element of G, and a maximal normal subgroup is a proper normal subgroup not contained in any larger proper normal subgroup. The cosocle of G, written Cosoc(G), is the intersection of all maximal normal subgroups: the elements common to every one of them. For example, for a quasisimple group the cosocle equals the center Z(G). The name comes from 'socle', French for pedestal, and the word is also used with related but different meanings for Lie algebras and modules.

 

In group theory the cosocle of a group G, denoted Cosoc(G), is defined as the intersection of all maximal normal subgroups of G. Since an intersection of normal subgroups is normal, Cosoc(G) is itself a normal subgroup. For a quasisimple group G, the cosocle coincides with the center, Cosoc(G) = Z(G). The term (from 'socle', meaning pedestal) has related meanings elsewhere: for a symmetric Lie algebra, the cosocle is the +1 eigenspace of its structural automorphism, and the algebra splits as a direct sum of socle and cosocle; in module theory, the cosocle of a module over a ring R is its maximal semisimple quotient. The catalog entry is anchored on the group-theoretic definition; carrying only the name or a loosely similar 'common part' idea into another setting does not count as an instance.

Scope of Application

  • Documented setting. In mathematics, the term cosocle (socle meaning pedestal in French) has several related meanings.

  • Documented setting. In group theory, a cosocle of a group G, denoted by Cosoc(G), is the intersection of all maximal normal subgroups of G.

  • Documented setting. In the context of Lie algebras, a cosocle of a symmetric Lie algebra is the eigenspace of its structural automorphism that corresponds to the eigenvalue +1.

  • Documented setting. (A symmetric Lie algebra decomposes into the direct sum of its socle and cosocle.).

  • Documented setting. In the context of module theory, the cosocle of a module over a ring R is defined to be the maximal semisimple quotient of the module.

Clarity

A clear use of Cosocle names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In group theory, a cosocle of a group G, denoted by Cosoc(G), is the intersection of all maximal normal subgroups of G.

Manages Complexity

Cosocle compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—in mathematics, the term cosocle (socle meaning pedestal in French) has several related meanings.—and the practical consequence—if G is a quasisimple group, then Cosoc(G) = Z(G). This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit.

Abstract Reasoning

  1. Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
  2. State the relation. Use the source-grounded identity: In group theory, a cosocle of a group G, denoted by Cosoc(G), is the intersection of all maximal normal subgroups of G.
  3. Check operation and conditions. In the context of Lie algebras, a cosocle of a symmetric Lie algebra is the eigenspace of its structural automorphism that corresponds to the eigenvalue +1.
  4. Demand recognition evidence.

Knowledge Transfer

Within the home domain. Knowledge about Cosocle transfers literally when a new case preserves the same carrier type, relation, and recognition test. In mathematics, the term cosocle (socle meaning pedestal in French) has several related meanings. In group theory, a cosocle of a group G, denoted by Cosoc(G), is the intersection of all maximal normal subgroups of G. Beyond the home domain. No canonical parent is asserted for Cosocle. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Relationships to Other Abstractions

Local relationship map for CosocleParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.CosocleDOMAINPrime abstraction: Group — is a kind ofGroupPRIME

Current abstraction Cosocle Domain-specific

Parents (1) — more general patterns this builds on

  • Cosocle is a kind of Group Prime

    A cosocle is an intersection of normal subgroups and therefore remains a group under the inherited operation.

Hierarchy paths (5) — routes to 5 parentless roots

Neighborhood in Abstraction Space

Cosocle sits in a sparse region of the domain-specific corpus (84th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Group-Theoretic Subgroup & Cohomology Structures (7 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08