Correspondence sentence (group theory)

In the mathematical sub-area of group theory, the correspondence theorem describes the fact that the subgroups in a factor group correspond exactly to those subgroups of the starting group that include the normal factor . The term correspondence set is used for similar relationships between substructures of other algebraic structures, albeit less often. ${\displaystyle G/N}$ ${\displaystyle N}$

Correspondence theorem in group theory

Let it be a surjective group homomorphism with a core . Then is the assignment ${\displaystyle \varphi :G\rightarrow H}$ ${\displaystyle N}$

${\displaystyle U\mapsto \varphi (U)}$

a bijection between the set of all comprehensive subgroups of onto the set of all subgroups of . ${\displaystyle N}$ ${\displaystyle U}$ ${\displaystyle G}$ ${\displaystyle H}$

${\displaystyle V\mapsto \varphi ^{-1}(V)}$

is the reverse image. The subgroups of thus correspond one-to-one to the subgroups of which contain. Normal dividers are mapped to normal dividers in both directions. ${\displaystyle H}$ ${\displaystyle G}$ ${\displaystyle N}$

If one specializes in this statement , one obtains that the subgroups (or normal divisors) are exactly those of the form with a subgroup (or normal divisor) . ${\displaystyle G/N\cong H}$ ${\displaystyle G/N}$ ${\displaystyle U/N}$ ${\displaystyle N\subset U\subset G}$

This mapping is monotone, ie for subgroups is true if and only if . ${\displaystyle N\subset U_{1},U_{2}\subset G}$ ${\displaystyle U_{1}\subset U_{2}}$ ${\displaystyle U_{1}/N\subset U_{2}/N}$

Conclusion : A normal divisor is exactly maximal among all normal divisors of , if is simple. ${\displaystyle N}$ ${\displaystyle G}$ ${\displaystyle G/N}$

Correspondence theorem in ring theory

Let it be a ring with a single element and a two-sided ideal. Then is the assignment ${\displaystyle R}$ ${\displaystyle {\mathfrak {a}}\subset R}$

${\displaystyle {\mathfrak {b}}\mapsto {\mathfrak {b}}/{\mathfrak {a}}}$

a bijection from the set of all comprehensive link ideals to the set of link ideals in This assignment is monotonic, that is, for link ideals , if and only if ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle R/{\mathfrak {a}}}$ ${\displaystyle {\mathfrak {a}}\subset {\mathfrak {b}},{\mathfrak {b'}}\subset R}$ ${\displaystyle {\mathfrak {b}}\subset {\mathfrak {b'}}}$ ${\displaystyle {\mathfrak {b}}/{\mathfrak {a}}\subset {\mathfrak {b'}}/{\mathfrak {a}}}$

Correspondence record for modules

Let there be a left- R module and a sub-module. Then is the assignment ${\displaystyle M}$ ${\displaystyle T\subset M}$

${\displaystyle S\mapsto S/T}$

a bijection of the set of all comprehensive sub-modules onto the set of all sub-modules of . This assignment is monotonous, i.e. it applies to sub-modules if and only if . ${\displaystyle T}$ ${\displaystyle S\subset M}$ ${\displaystyle M/T}$ ${\displaystyle T\subset S,S'\subset M}$ ${\displaystyle S\subset S'}$ ${\displaystyle S/T\subset S'/T}$

Individual evidence

1. ↑ Christian Karpfinger: Algebra, Groups - Rings - Body , Spectrum of Science (2013), ISBN 978-3-8274-3011-3 , sentence 4.13 (correspondence sentence)
2. ↑ DJS Robinson : A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , Chapter 1.4, page 20, Subgroups of the Image
3. ↑ Christian Karpfinger: Algebra, Groups - Rings - Body , Spectrum of Science (2013), ISBN 978-3-8274-3011-3 , Lemma 11.2
4. ^ Joseph J. Rotman: An Introduction to Homological Algebra , Academic Press Inc. (1979), ISBN 978-0-12-599250-3 , Sentence 2.15 (Correspondence Theorem for Rings)
5. ↑ Gerd Fischer: Textbook of Algebra , Gabler-Verlag (2013), ISBN 978-3-658-02220-4 , Chapter II.2.4, correspondence set for ideals
6. ↑ Christian Karpfinger: Algebra, Groups - Rings - Body , Spectrum of Science (2013), ISBN 978-3-8274-3011-3 , sentence 15.14 (correspondence sentence)
7. ^ Joseph J. Rotman: An Introduction to Homological Algebra , Academic Press Inc. (1979), ISBN 978-0-12-599250-3 , Theorem 2.14 (Correspondence Theorem)