Socle (mathematics)#Socle of a module

In the context of group theory, the socle of a group G, denoted soc(G), is the subgroup generated by the minimal normal subgroups of G. It can happen that a group has no minimal non-trivial normal subgroup (that is, every non-trivial normal subgroup properly contains another such subgroup) and in that case the socle is defined to be the subgroup generated by the identity. The socle is a direct product of minimal normal subgroups.{{sfn|Robinson|1996|loc=p.87}}

As an example, consider the cyclic group Z₁₂ with generator u, which has two minimal normal subgroups, one generated by u⁴ (which gives a normal subgroup with 3 elements) and the other by u⁶ (which gives a normal subgroup with 2 elements). Thus the socle of Z₁₂ is the group generated by u⁴ and u⁶, which is just the group generated by u².

The socle is a characteristic subgroup, and hence a normal subgroup. It is not necessarily transitively normal, however.

If a group G is a finite solvable group, then the socle can be expressed as a product of elementary abelian p-groups. Thus, in this case, it is just a product of copies of Z/pZ for various p, where the same p may occur multiple times in the product.

In the context of module theory and ring theory the socle of a module M over a ring R is defined to be the sum of the minimal nonzero submodules of M. It can be considered as a dual notion to that of the radical of a module. In set notation,

:$\backslash mathrm\{soc\}(M)\; =\; \backslash sum\_\{N\; \backslash text\{\; is\; a\; simple\; submodule\; of\; \}M\}\; N.$

Equivalently,

:$\backslash mathrm\{soc\}(M)\; =\; \backslash bigcap\_\{E\; \backslash text\{\; is\; an\; essential\; submodule\; of\; \}M\}\; E.$

The socle of a ring R can refer to one of two sets in the ring. Considering R as a right R-module, soc(R_R) is defined, and considering R as a left R-module, soc(_RR) is defined. Both of these socles are ring ideals, and it is known they are not necessarily equal.

• If M is an Artinian module, soc(M) is itself an essential submodule of M.

In fact, if M is a semiartinian module, then soc(M) is itself an essential submodule of M. Additionally, if M is a non-zero module over a left semi-Artinian ring, then soc(M) is itself an essential submodule of M. This is because any non-zero module over a left semi-Artinian ring is a semiartinian module.

• A module is semisimple if and only if soc(M) = M. Rings for which soc(M) = M for all M are precisely semisimple rings.
• soc(soc(M)) = soc(M).
• M is a finitely cogenerated module if and only if soc(M) is finitely generated and soc(M) is an essential submodule of M.
• Since the sum of semisimple modules is semisimple, the socle of a module could also be defined as the unique maximal semisimple submodule.
• From the definition of rad(R), it is easy to see that rad(R) annihilates soc(R). If R is a finite-dimensional unital algebra and M a finitely generated R-module then the socle consists precisely of the elements annihilated by the Jacobson radical of R.J. L. Alperin; Rowen B. Bell, Groups and Representations, 1995, {{isbn|0-387-94526-1}}, p. 136

In the context of Lie algebras, a socle 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]

• Injective hull
• Radical of a module
• Cosocle

{{reflist}}

• {{cite book | last1 = Alperin | first1 = J.L. | author-link = J. L. Alperin | last2= Bell | first2=Rowen B. | title = Groups and Representations | publisher = Springer-Verlag | year = 1995| url =https://archive.org/details/groupsrepresenta00jlal| url-access = limited | isbn = 0-387-94526-1 | page = [https://archive.org/details/groupsrepresenta00jlal/page/n146 136]}}
• {{cite book | last1=Anderson | first1=Frank Wylie | last2=Fuller|first2=Kent R. | title=Rings and Categories of Modules | publisher=Springer-Verlag | isbn=978-0-387-97845-1 | year=1992}}
• {{citation |last1=Robinson |first1=Derek J. S. |title=A course in the theory of groups |series=Graduate Texts in Mathematics |volume=80 |edition=2 |publisher=Springer-Verlag |place=New York |year=1996 |pages=xviii+499 |isbn=0-387-94461-3 |mr=1357169 |doi=10.1007/978-1-4419-8594-1}}

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Category:Module theory

Category:Group theory

Category:Functional subgroups