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.[1] If G 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]

## See also

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* Socle
* Radical of a module

## References

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1. ^ a b Adolfo Ballester-Bolinches, Luis M. Ezquerro, Classes of Finite Groups, 2006, ISBN 1402047185, p. 97
2. ^ Mikhail Postnikov, Geometry VI: Riemannian Geometry, 2001, ISBN 3540411089,p. 98
3. ^ Braden, Tom; Licata, Anthony; Phan, Christopher; Proudfoot, Nicholas; Webster, Ben (2011). "Localization algebras and deformations of Koszul algebras". Selecta Math. 17 (3): 533–572\. arXiv:0905.1335. doi:10.1007/s00029-011-0058-y. S2CID 16184908. "Lemma 3.8"

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