First isomorphism theorem: Difference between revisions
Created page with "The '''first isomorphism theorem''' is a fundamental result in abstract algebra that describes the relationship between a homomorphism, its kernel, and its image. The theorem appears uniformly across algebraic structures such as groups, rings, and modules, and serves as a prototype for many structural results in algebra. Specifically, given a homeo..." |
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* The image of <math>f</math> is a subgroup of <math>H</math>. | * The image of <math>f</math> is a subgroup of <math>H</math>. | ||
* <math>G/\ker f \cong \operatorname{im} f</math> . | * <math>G/\ker f \cong \operatorname{im} f</math>. | ||
Revision as of 19:19, 4 April 2026
The first isomorphism theorem is a fundamental result in abstract algebra that describes the relationship between a homomorphism, its kernel, and its image. The theorem appears uniformly across algebraic structures such as groups, rings, and modules, and serves as a prototype for many structural results in algebra. Specifically, given a homeomorphism, the quotient of its domain by its kernel is isomorphic to its image.
Group theory
Statement
Let 
and 
be groups and 
a group homomorphism. Then,
- The kernel of
is a normal subgroup of . - The image of is a subgroup of .
.