“Correspondence theorem (group theory)”的意思、由来-开放百科全书

In the area of mathematics known as group theory, the correspondence theorem,^[1]^[2]^[3]^[4]^[5]^[6]^[7]^[8] sometimes referred to as the fourth isomorphism theorem^[6]^[9]^[10]^[11] or the lattice theorem,^[12] states that if

is a normal subgroup of a group

, then there exists a bijection from the set of all subgroups

containing

, onto the set of all subgroups of the quotient group

. The structure of the subgroups of

is exactly the same as the structure of the subgroups of

containing

, with

collapsed to the identity element.

This list is far from exhaustive. In fact, most properties of subgroups are preserved in their images under the bijection onto subgroups of a quotient group.

More generally, there is a monotone Galois connection

between the lattice of subgroups of

(not necessarily containing

) and the lattice of subgroups of

: the lower adjoint of a subgroup

is given by

and the upper adjoint of a subgroup

is a given by

. The associated closure operator on subgroups of

; the associated kernel operator on subgroups of

is the identity.

See also

Notes

1. ^{{cite book|author=Derek John Scott Robinson|title=An Introduction to Abstract Algebra|year=2003|publisher=Walter de Gruyter|isbn=978-3-11-017544-8|page=64}}
2. ^{{cite book|author=J. F. Humphreys|title=A Course in Group Theory|year=1996|publisher=Oxford University Press|isbn=978-0-19-853459-4|page=65}}
3. ^{{cite book|author=H.E. Rose|title=A Course on Finite Groups|year=2009|publisher=Springer|isbn=978-1-84882-889-6|page=78}}
4. ^{{cite book|author1=J.L. Alperin|author2=Rowen B. Bell|title=Groups and Representations|year=1995|publisher=Springer|isbn=978-1-4612-0799-3|page=11}}
5. ^{{cite book|author=I. Martin Isaacs|title=Algebra: A Graduate Course|year=1994|publisher=American Mathematical Soc.|isbn=978-0-8218-4799-2|page=35}}
6. ^¹{{cite book|author=Joseph Rotman|title=An Introduction to the Theory of Groups|year=1995|publisher=Springer|isbn=978-1-4612-4176-8|pages=37–38|edition=4th}}
7. ^{{cite book|author=W. Keith Nicholson|title=Introduction to Abstract Algebra|year=2012|publisher=John Wiley & Sons|isbn=978-1-118-31173-8|page=352|edition=4th}}
8. ^{{cite book|author=Steven Roman|title=Fundamentals of Group Theory: An Advanced Approach|year=2011|publisher=Springer Science & Business Media|isbn=978-0-8176-8301-6|pages=113–115}}
9. ^{{cite book|author1=Jonathan K. Hodge|author2=Steven Schlicker|author3=Ted Sundstrom|title=Abstract Algebra: An Inquiry Based Approach|year=2013|publisher=CRC Press|isbn=978-1-4665-6708-5|page=425}}
10. ^Some authors use "fourth isomorphism theorem" to designate the Zassenhaus lemma; see for example by Alperin & Bell (p. 13) or {{cite book|author=Robert Wilson|title=The Finite Simple Groups|year=2009|publisher=Springer|isbn=978-1-84800-988-2|page=7}}
11. ^Depending how one counts the isomorphism theorems, the correspondence theorem can also be called the 3rd isomorphism theorem; see for instance H.E. Rose (2009), p. 78.
12. ^W.R. Scott: Group Theory, Prentice Hall, 1964, p. 27.

See also

Notes

References