Homomorphism theorem

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The homomorphism theorem is a mathematical theorem from the field of algebra , which applies in a corresponding form to mappings between groups , vector spaces and rings . It creates a close connection between group homomorphisms and normal divisors , vector space homomorphisms and sub-vector spaces as well as ring homomorphisms and ideals . The homomorphism theorem reads:

Is a homomorphism and the core by , then the quotient is isomorphic to the image .



If a group is homomorphic , then the kernel is a normal divisor of and the factor group is isomorphic to the image . A corresponding isomorphism is given by .


It suffices to show that the map is a group isomorphism .

is well defined and injective , there

is a group homomorphism , since for all secondary classes and the following applies:

surjective because for each applies: .

It follows from this that is a group isomorphism , and thus .


is a group homomorphism whose core consists of the special linear group of matrices with determinants . According to the homomorphism theorem applies
From this it follows in particular that, in contrast to the linear group, the factor group is Abelian.
  • Analogously one shows:
where stands for the orthogonal group and for the special orthogonal group.
  • It stands for the symmetrical group . The Signum mapping defines a group homomorphism with ( alternating group ), which is for surjective. According to the homomorphism theorem, we have :


If a ring homomorphism , then the kernel is an ideal of and the factor ring is isomorphic to the image .

The proof is analogous to the proof for groups , it only needs to be shown:

Vector space


Is a vector space homomorphism, i. H. a linear mapping from to , then the kernel is a subspace of and the factor space is isomorphic to the image .


The differential operator

is a homomorphism from the vector space of the continuously differentiable functions into the vector space of the continuous functions . Its core is the set of constant functions that is noted here as . According to the homomorphism theorem applies

The isomorphism is the induced homomorphism


Its inverse homomorphism is indefinite integration

where is any antiderivative of .


  • Homomorphism theorem for algebraic structures:
If and are two algebraic structures of the same kind and if a homomorphism of this kind with a kernel , then applies .
  • The theorem applies generally to every Abelian category .
  • The theorem also applies to the category of topological groups , for example ; however, the picture is then also to be understood in the categorical sense, so it is generally not the set-theoretical picture with the induced topology . Also, a bijective continuous homomorphism is only a categorical isomorphism if its inverse is continuous, i.e. H. if it is also a homeomorphism .


  • Christian Karpfinger, Kurt Meyberg: Algebra. Groups - rings - bodies. Spektrum Akademischer Verlag, Heidelberg 2009, ISBN 978-3-8274-2018-3 , p. 54, p. 167-168