cosocle

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. Adolfo Ballester-Bolinches, Luis M. Ezquerro, Classes of Finite Groups, 2006, {{ISBN|1402047185}}, [https://books.google.com/books?id=VoQ53SosWLIC&dq=cosocle&pg=PA97 p. 97] If G is a quasisimple group, then Cosoc(G) = Z(G).

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.)Mikhail Postnikov, Geometry VI: Riemannian Geometry, 2001, {{ISBN|3540411089}},[https://books.google.com/books?id=P60o2UKOaPcC&q=socle&pg=PA98 p. 98]

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.{{cite journal |last1= Braden|first1= Tom|last2=Licata|first2=Anthony|last3=Phan|first3=Christopher|last4=Proudfoot|first4=Nicholas|last5=Webster|first5=Ben|date=2011|title=Localization algebras and deformations of Koszul algebras|journal=Selecta Math.|volume=17|issue=3|pages=533–572|doi=10.1007/s00029-011-0058-y|quote=Lemma 3.8|arxiv=0905.1335|s2cid= 16184908}}