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

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