TOPICS
Search

First Group Isomorphism Theorem


The first group isomorphism theorem, also known as the fundamental homomorphism theorem, states that if phi:G->H is a group homomorphism, then Ker(phi)⊴G and G/Ker(phi)=phi(G), where N⊴G indicates that N is a normal subgroup of G, Ker(G) denotes the group kernel, and G=H indicates that G and H are isomorphic groups.

A corollary states that if phi:G->H is a group homomorphism, then

1. phi is injective iff Ker(phi)=e_G

2. |G:Ker(phi)|=|phi(G)|, where |G| denotes the group order of a group G.


See also

Second Group Isomorphism Theorem, Third Group Isomorphism Theorem, Fourth Group Isomorphism Theorem

This entry contributed by Nick Hutzler

Explore with Wolfram|Alpha

References

Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 98-100, 1998.

Referenced on Wolfram|Alpha

First Group Isomorphism Theorem

Cite this as:

Hutzler, Nick. "First Group Isomorphism Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/FirstGroupIsomorphismTheorem.html

Subject classifications