You are here
Homesecond isomorphism theorem
Primary tabs
second isomorphism theorem
Let $(G,*)$ be a group. Let $H$ be a subgroup of $G$ and let $K$ be a normal subgroup of $G$. Then

$HK:=\{h*k\mid h\in H,\ k\in K\}$ is a subgroup of $G$,

$K$ is a normal subgroup of $HK$,

$H\cap K$ is a normal subgroup of $H$,

There is a natural group isomorphism $H/(H\cap K)=HK/K$.
Type of Math Object:
Theorem
Major Section:
Reference
Groups audience:
Mathematics Subject Classification
13C99 no label found20A05 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
Recent Activity
Apr 20
new image: informationtheoreticdistributedmeasurementdds.png by rspuzio
new image: informationtheoreticdistributedmeasurement4.2 by rspuzio
new image: informationtheoreticdistributedmeasurement4.1 by rspuzio
new image: informationtheoreticdistributedmeasurement3.2 by rspuzio
new image: informationtheoreticdistributedmeasurement3.1 by rspuzio
new image: informationtheoreticdistributedmeasurement2.1 by rspuzio
Apr 19
new collection: On the InformationTheoretic Structure of Distributed Measurements by rspuzio
Apr 15
new question: Prove a formula is part of the Gentzen System by LadyAnne
Mar 30
new question: A problem about Euler's totient function by mbhatia
new problem: Problem: Show that phi(a^n1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia
new image: informationtheoreticdistributedmeasurementdds.png by rspuzio
new image: informationtheoreticdistributedmeasurement4.2 by rspuzio
new image: informationtheoreticdistributedmeasurement4.1 by rspuzio
new image: informationtheoreticdistributedmeasurement3.2 by rspuzio
new image: informationtheoreticdistributedmeasurement3.1 by rspuzio
new image: informationtheoreticdistributedmeasurement2.1 by rspuzio
Apr 19
new collection: On the InformationTheoretic Structure of Distributed Measurements by rspuzio
Apr 15
new question: Prove a formula is part of the Gentzen System by LadyAnne
Mar 30
new question: A problem about Euler's totient function by mbhatia
new problem: Problem: Show that phi(a^n1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia