Fourth isomorphism theorem
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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This result is termed the lattice isomorphism theorem, the fourth isomorphism theorem, and the correspondence theorem.
Statement with symbols
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 .
- 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)