Cosocle
Encyclopedia
In mathematics
, the term cosocle has several related meanings.
In group theory
, a cosocle of a group
G, denoted by Cosoc(G), is the intersection of all maximal
normal subgroup
s of G.
If G is a quasisimple group
, then Cosoc(G) = Z(G)
.
In the context of Lie algebra
s, a cosocle of a symmetric Lie algebra is the eigenspace of its structural automorphism
which corresponds to the eigenvalue +1. (A symmetric Lie algebra decomposes into the direct sum
of its socle
and cosocle.)
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the term cosocle has several related meanings.
In group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, a cosocle of a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G, denoted by Cosoc(G), is the intersection of all maximal
Maximal subgroup
In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra.In group theory, a maximal subgroup H of a group G is a proper subgroup, such that no proper subgroup K contains H strictly. In other words H is a maximal element of the partially...
normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
s of G.
If G is a quasisimple group
Quasisimple group
In mathematics, a quasisimple group is a group that is a perfect central extension E of a simple group S...
, then Cosoc(G) = Z(G)
Center (group theory)
In abstract algebra, the center of a group G, denoted Z,The notation Z is from German Zentrum, meaning "center". is the set of elements that commute with every element of G. In set-builder notation,...
.
In the context of Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
s, a cosocle of a symmetric Lie algebra is the eigenspace of its structural automorphism
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
which corresponds to the eigenvalue +1. (A symmetric Lie algebra decomposes into the direct sum
Direct sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as submodules...
of its socle
Socle (mathematics)
-Socle of a group:In the context of group theory, the socle of a group G, denoted Soc, is the subgroup generated by the minimal non-trivial normal subgroups of G. The socle is a direct product of minimal normal subgroups...
and cosocle.)