Fourth isomorphism theorem
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This article gives the statement, and possibly proof, of a basic fact in group theory.
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This article is about an isomorphism theorem in group theory.
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Name
This result is termed the lattice isomorphism theorem, the fourth isomorphism theorem, and the correspondence theorem.
Statement
Statement with symbols
Let be a group and let be a Normal subgroup (?) of . Then, we have a bijection:
Set of subgroups of containing Set of subgroups of
If is the quotient map, then this bijection is given by:
in the forward direction, and:
in the reverse direction. Moreover:
- Under the bijection, normality is preserved. In other words, a subgroup containing is normal if and only if its image under is normal.
- The bijection is an isomorphism between the lattice of subgroups of containing , and the lattice of subgroups of . In other words, the bijection preserves partial order: if and only if . It also preserves intersections and joins.
- The bijection preserves index. If are subgroups of containing , with , then .
References
Textbook references
- Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 99, Theorem 20, Section 3.3 (few steps of proof given, but full proof not provided)
- Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, ^{More info}, Page 75, Exercise 8, Section 7 (Restriction of a homomorphism to a subgroup) (starred problem, termed Correspondence Problem)