second isomorphism theorem

Let (G,*) be a group. Let H be a subgroupMathworldPlanetmathPlanetmath of G and let K be a normal subgroupMathworldPlanetmath of G. Then

  • HK:={h*khH,kK} is a subgroup of G,

  • K is a normal subgroup of HK,

  • HK is a normal subgroup of H,

  • There is a natural group isomorphism H/(HK)=HK/K.

The same statement also holds in the categoryMathworldPlanetmath of modules over a fixed ring (where normality is neither needed nor relevant), and indeed can be formulated so as to hold in any abelian categoryMathworldPlanetmathPlanetmathPlanetmath.

Title second isomorphism theorem
Canonical name SecondIsomorphismTheorem
Date of creation 2013-03-22 12:08:46
Last modified on 2013-03-22 12:08:46
Owner djao (24)
Last modified by djao (24)
Numerical id 9
Author djao (24)
Entry type Theorem
Classification msc 13C99
Classification msc 20A05