Homomorphism theorem

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 . ${\displaystyle f\colon A\to B}$ ${\displaystyle \ker(f)}$ ${\displaystyle f}$ ${\displaystyle A/\ker(f)}$ ${\displaystyle f(A)}$

group

statement

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 . ${\displaystyle f\colon \left(G,\circ \right)\to \left(H,\star \right)}$ ${\displaystyle N\colon =\ker \left(f\right)}$ ${\displaystyle G}$ ${\displaystyle G/N}$ ${\displaystyle f\left(G\right)}$ ${\displaystyle {\tilde {f}}\colon G/N\rightarrow f(G);gN\mapsto f\left(g\right)}$

proof

It suffices to show that the map is a group isomorphism . ${\displaystyle {\tilde {f}}}$

${\displaystyle {\tilde {f}}}$is well defined and injective , there

${\displaystyle aN=bN\Leftrightarrow b^{-1}a\in N\Leftrightarrow f(b^{-1}a)=e\Leftrightarrow {\tilde {f}}(aN)=f(a)=f(b)={\tilde {f}}(bN)}$

${\displaystyle {\tilde {f}}}$is a group homomorphism , since for all secondary classes and the following applies: ${\displaystyle aN}$ ${\displaystyle bN}$

${\displaystyle {\tilde {f}}\left(aN\circ bN\right)={\tilde {f}}\left(abN\right)=f(ab)=f(a)\star f(b)={\tilde {f}}(aN)\star {\tilde {f}}(bN)}$

${\displaystyle {\tilde {f}}}$ surjective because for each applies: . ${\displaystyle g\colon =f\left(g'\right)\in f\left(G\right)}$ ${\displaystyle {\tilde {f}}\left(g'N\right)=f\left(g'\right)=g}$

It follows from this that is a group isomorphism , and thus . ${\displaystyle {\tilde {f}}\colon G/N\rightarrow f(G)}$ ${\displaystyle G/N\cong f\left(G\right)}$

Examples

• Let it stand for the general linear group represented by regular matrices over a body . The determinant ${\displaystyle \operatorname {GL} (n,K)}$ ${\displaystyle K}$

${\displaystyle \det \colon \operatorname {GL} (n,K)\to K^{*}=K\setminus \{0\}}$

is a group homomorphism whose core consists of the special linear group of matrices with determinants . According to the homomorphism theorem applies ${\displaystyle \operatorname {SL} (n,K)}$ ${\displaystyle n\times n}$ ${\displaystyle 1}$

${\displaystyle \operatorname {GL} (n,K)/\operatorname {SL} (n,K)\cong K^{*}}$.

From this it follows in particular that, in contrast to the linear group, the factor group is Abelian. ${\displaystyle \operatorname {GL} (n,K)}$ ${\displaystyle \operatorname {GL} (n,K)/\operatorname {SL} (n,K)}$

• Analogously one shows:

${\displaystyle \operatorname {O} (n,K)/\operatorname {SO} (n,K)\cong \left\{-1,1\right\}}$

where stands for the orthogonal group and for the special orthogonal group. ${\displaystyle \operatorname {O} (n,K)}$ ${\displaystyle \operatorname {SO} (n,K)}$

• 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 : ${\displaystyle S_{n}}$ ${\displaystyle \operatorname {sign} \colon S_{n}\to \left\{-1,1\right\}}$ ${\displaystyle \operatorname {ker} \left(\operatorname {sign} \right)=\operatorname {Alt} _{n}}$ ${\displaystyle n\geq 2}$ ${\displaystyle n\geq 2}$

  ${\displaystyle S_{n}/\operatorname {Alt} _{n}\cong \left\{-1,1\right\}}$

ring

If a ring homomorphism , then the kernel is an ideal of and the factor ring is isomorphic to the image . ${\displaystyle f\colon R\to S}$ ${\displaystyle \ker(f)}$ ${\displaystyle R}$ ${\displaystyle R/{\ker(f)}}$ ${\displaystyle \operatorname {im} (f)}$

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

${\displaystyle {\tilde {f}}\left(aN\cdot bN\right)={\tilde {f}}\left(\left(a\cdot b\right)N\right)=f\left(a\cdot b\right)=f\left(a\right)\cdot f\left(b\right)={\tilde {f}}\left(aN\right)\cdot {\tilde {f}}\left(bN\right)}$

Vector space

statement

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 . ${\displaystyle f}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle \ker(f)}$ ${\displaystyle V}$ ${\displaystyle V/{\ker(f)}}$ ${\displaystyle \operatorname {im} (f)}$

example

The differential operator

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\colon \ C^{1}(\mathbb {R} )\rightarrow C^{0}(\mathbb {R} ),\quad f(x)\mapsto {\frac {\mathrm {d} }{\mathrm {d} x}}f(x)=f'(x)}$

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 ${\displaystyle \mathbb {R} }$ ${\displaystyle C^{1}(\mathbb {R} )}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle C^{0}(\mathbb {R} )}$ ${\displaystyle \mathbb {R} }$

${\displaystyle C^{1}(\mathbb {R} )/\mathbb {R} \cong C^{0}(\mathbb {R} )}$

The isomorphism is the induced homomorphism

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}^{\displaystyle {\tilde {}}}:C^{1}(\mathbb {R} )/\mathbb {R} \rightarrow C^{0}(\mathbb {R} ),\quad f(x)+\mathbb {R} \mapsto f'(x)}$.

Its inverse homomorphism is indefinite integration

${\displaystyle \int \cdot \,\,\mathrm {d} x\colon \ C^{0}(\mathbb {R} )\rightarrow C^{1}(\mathbb {R} )/\mathbb {R} ,\quad g(x)\mapsto \int g(x){\mathrm {d} }x=G(x)+\mathbb {R} ,}$

where is any antiderivative of . ${\displaystyle G(x)}$ ${\displaystyle g(x)}$

Generalizations

• 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 . ${\displaystyle (A,(f_{i})_{i\in \{1,\dotsc ,n\}})}$ ${\displaystyle (B,(g_{i})_{i\in \{1,\dotsc ,n\}})}$ ${\displaystyle \varphi \colon A\to B}$ ${\displaystyle \theta _{\varphi }}$ ${\displaystyle A/\theta _{\varphi }\simeq \varphi (A)}$

• 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 .

literature

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