fourth isomorphism theorem

Theorem 1 (The Fourth Isomorphism Theorem)

Let G be a group and NG. There is a bijection between G(N), the set of subgroupsMathworldPlanetmathPlanetmath of G containing N, and the set of subgroups of G/N defined by AA/N. Moreover, for any two subgroups A,B in G(N), we have

  1. 1.

    AB if and only if A/NB/N,

  2. 2.

    AB implies |B:A|=|B/N:A/N|,

  3. 3.


  4. 4.

    (AB)/N=(A/N)(B/N), and

  5. 5.

    AG if and only if (A/N)(G/N).

Title fourth isomorphism theorem
Canonical name FourthIsomorphismTheorem
Date of creation 2013-03-22 14:00:52
Last modified on 2013-03-22 14:00:52
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 16
Author bwebste (988)
Entry type Theorem
Classification msc 20A05
Synonym correspondence theorem
Synonym lattice isomorphism theorem