The term cap has several related meanings in mathematics .
Content
Group
In the context of group theory, the socle of G , denoted by soc ( G ), is the subgroup generated by the of G. It may happen that a group does not have a minimal nontrivial normal subgroup (that is, any nontrivial normal subgroup contains another such subgroup), in this case the socle is defined as the subgroup generated by the unit element. A socle is a direct product of characteristic simple groups [1] .
As an example, consider a cyclic group Z 12 with generator u that has two minimal normal subgroups, one generated by element u 4 (which gives a normal subgroup with 3 elements), and the other by element u 6 (which gives a normal subgroup with 2 elements). Then the base of the group Z 12 is the group generated by the elements u 4 and u 6 , which is simply generated by the element u 2 .
The base is a characteristic subgroup , and therefore a normal subgroup. She, however, is not necessarily .
If the group G is a finite solvable group , then the socle can be expressed as the product of p-groups . In this case, it is simply the product of copies of Z / pZ for different p , where some p can occur several times.
Module Base
In the context of the module over the ring and ring theory, the base of the module M over the ring R is defined as the sum of the minimal nonzero submodules of the module M. It can be considered as dual for the . In the notation of set theory
- where the summation is carried out over all submodules of the module M
Which is equivalent
- , where the intersection is carried out over all essential submodules of the module M
The base of the ring R may belong to one of the sets in the ring. Suppose a right module R , soc ( R R ), is defined, and a left module, soc ( R R ), is defined. Both of these socles are the ideals of the rings and it is known that they do not necessarily coincide.
- If M is an Artinian module , soc ( M ) is itself an module M.
- A module is semisimple if and only if soc ( M ) = M. Rings for which soc ( M ) = M for all M are exactly semisimple modules .
- soc (soc ( M )) = soc ( M ).
- M is a finitely generated module if and only if soc ( M ) is finitely generated and soc (M) is an of M.
- Since the sum of semisimple modules is a semisimple module, the module base can be defined as the only maximal semisimple submodule.
- From the definition of rad ( R ) it is easy to see that rad ( R ) soc ( R ). If R is a finite-dimensional unital algebra and M is a finitely generated R -module, then the base consists of exactly the elements annihilated by the Johnson radical of the ring R [2] .
The Lie Algebra Base
In the context of Lie algebras, the base of a is the eigenspace of its structural automorphisms that correspond to the eigenvalue −1. (The symmetric Lie algebra is divided into the direct sum of its socle and .) [3] .
See also
- Injective sheath
Notes
- ↑ Robinson, 1996 , p. 87.
- ↑ Alperin, Bell, 1995 , p. 136.
- ↑ Postnikov, 2001 , p. 98.
Literature
- Alperin JL, Bell RB Groups and Representations. - Springer-Verlag , 1995 .-- S. 136. - ISBN 0-387-94526-1 .
- Frank Wylie Anderson, Kent R. Fuller. Rings and Categories of Modules. - Springer-Verlag , 1992. - ISBN 978-0-387-97845-1 .
- Derek JS Robinson. A course in the theory of groups. - 2. - New York: Springer-Verlag , 1996.- T. 80.- C. xviii + 499. - ( Graduate Texts in Mathematics ). - ISBN 0-387-94461-3 . - DOI : 10.1007 / 978-1-4419-8594-1 .
- Mikhail Postnikov . Geometry VI: Riemannian Geometry . - 2001. - ISBN 3540411089 .
- Postnikov M.M. Lecture 8, Symmetric algebras and Lie ternars. // Riemannian geometry Semester V. - Moscow: Factorial, 1998. - (Lectures on geometry). - ISBN 5-88688-020-8 .
- Alperin JL, Bell RB Groups and Representations. - 1995. - T. 162. - (Graduate Texts in Mathematics). - ISBN 0-387-94526-1 .