Inmathematics, the term cosocle (socle meaning pedestalinFrench) has several related meanings.
Ingroup theory, a cosocle of a group G, denoted by Cosoc(G), is the intersection of all maximal normal subgroupsofG.[1]IfG is a quasisimple group, then Cosoc(G) = Z(G).[1]
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. (A symmetric Lie algebra decomposes into the direct sum of its socle and cosocle.)[2]
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.[3]
Lemma 3.8
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