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 G be a group and let N be a Normal subgroup (?) of G. Then, we have a bijection:

Set of subgroups of G containing N  \leftrightarrow Set of subgroups of G/N

If \varphi:G \to G/N is the quotient map, then this bijection is given by:

H \mapsto \varphi(H)

in the forward direction, and:

K \mapsto \varphi^{-1}(K)

in the reverse direction. Moreover:

  1. Under the bijection, normality is preserved. In other words, a subgroup containing N is normal if and only if its image under \varphi is normal.
  2. The bijection is an isomorphism between the lattice of subgroups of G containing N, and the lattice of subgroups of G/N. In other words, the bijection preserves partial order: A \le B if and only if \varphi(A) \le \varphi(B). It also preserves intersections and joins.
  3. The bijection preserves index. If A,B are subgroups of G containing N, with A \le B, then [B:A] = [\varphi(B):\varphi(A)].

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)