Difference between revisions of "First group isomorphism theorem/Statement"

From Maths
Jump to: navigation, search
(Saving work)
 
m (Alec moved page First group isomorphism theorem\Statement to First group isomorphism theorem/Statement without leaving a redirect: Typo)
 
(No difference)

Latest revision as of 19:34, 15 July 2016

Statement

Let [ilmath](G,*)[/ilmath] and [ilmath](H,*)[/ilmath] be groups. Let [ilmath]\varphi:G\rightarrow H[/ilmath] be a group homomorphism, then[1]:

  • [ilmath]G/\text{Ker}(\varphi)\cong\text{Im}(\varphi)[/ilmath]
    • Explicitly we may state this as: there exists a group isomorphism between [ilmath]G/\text{Ker}(\varphi)[/ilmath] and [ilmath]\text{Im}(\varphi)[/ilmath].

Note: the special case of [ilmath]\varphi[/ilmath] being surjective, then [ilmath]\text{Im}(\varphi)=H[/ilmath], so we see [ilmath]G/\text{Ker}(\varphi)\cong H[/ilmath]

References

  1. Abstract Algebra - Pierre Antoine Grillet