Group homomorphism

In group theory , one looks at special mappings between groups , which are called group homomorphisms. A group homomorphism is a mapping between two groups that is compatible with them , and thus a special homomorphism .

definition

Two groups are given and a function is called group homomorphism if the following applies to all elements : ${\ displaystyle (G, *)}$${\ displaystyle (H, \ star).}$${\ displaystyle \ phi \ colon G \ to H}$${\ displaystyle g_ {1}, g_ {2} \ in G}$

${\ displaystyle \ phi (g_ {1} * g_ {2}) = \ phi (g_ {1}) \ star \ phi (g_ {2}).}$

The equation says that the homomorphism is structure-preserving : It does not matter whether you first connect two elements and display the result or whether you first display the two elements and then connect the images.

From this definition it follows that a group homomorphism maps the neutral element of to the neutral element of : ${\ displaystyle e_ {G}}$${\ displaystyle G}$${\ displaystyle e_ {H}}$${\ displaystyle H}$

${\ displaystyle \ phi (e_ {G}) = e_ {H},}$

because for all true ${\ displaystyle g \ in G}$

${\ displaystyle \ phi (g) = \ phi (g * e_ {G}) = \ phi (g) \ star \ phi (e_ {G}),}$

so the neutral element is in . ${\ displaystyle \ phi (e_ {G})}$${\ displaystyle H}$

It also follows that it maps inverse to inverse:

${\ displaystyle \ phi (g ^ {- 1}) = \ phi (g) ^ {- 1}}$ for all ${\ displaystyle g \ in G,}$

because because of

${\ displaystyle e_ {H} = \ phi (e_ {G}) = \ phi (g * g ^ {- 1}) = \ phi (g) \ star \ phi (g ^ {- 1})}$

is the inverse of${\ displaystyle \ phi (g ^ {- 1})}$${\ displaystyle \ phi (g).}$

Image and core

As picture (Engl. Image ) of the homomorphism is called the image set from among : ${\ displaystyle f \ colon G \ to H}$${\ displaystyle G}$${\ displaystyle f}$

${\ displaystyle f (G) = \ operatorname {image} (f) = \ operatorname {im} (f) \ colon = \ left \ {f (u) \ mid u \ in G \ right \}}$

The core (engl. Kernel ) of the prototype of the neutral element : ${\ displaystyle f}$${\ displaystyle e_ {H}}$

${\ displaystyle f ^ {- 1} (e_ {H}) = \ operatorname {Kern} (f) = \ operatorname {ker} (f) \ colon = \ left \ {u \ in G \ mid f (u) = e_ {H} \ right \}}$

Exactly when applies (the core of thus only contains the neutral element of , which is always in the core), is injective . An injective group homomorphism is also called group monomorphism . ${\ displaystyle \ operatorname {core} (f) = \ left \ {e_ {G} \ right \}}$${\ displaystyle f}$${\ displaystyle G}$${\ displaystyle f}$

The core of is always a normal subgroup of and the image of is a subgroup of . According to the theorem of homomorphism , the factor group is isomorphic to . ${\ displaystyle f}$${\ displaystyle G}$${\ displaystyle f}$${\ displaystyle H}$ ${\ displaystyle G / \ operatorname {core} (f)}$${\ displaystyle \ operatorname {image} (f)}$

Examples

Trivial examples

• If and are arbitrary groups, then the mapping that maps each element to the neutral element of is a group homomorphism. Its core is whole .${\ displaystyle G}$${\ displaystyle H}$${\ displaystyle h \ colon G \ to H}$${\ displaystyle H}$${\ displaystyle G}$
• For each group is the identical mapping , a bijective group homomorphism.${\ displaystyle G}$ ${\ displaystyle \ operatorname {id} \ colon G \ to G, \ \ operatorname {id} (x) = x}$
• If the group is a subgroup , the inclusion map is an injective group homomorphism of in .${\ displaystyle H}$${\ displaystyle G}$ ${\ displaystyle i \ colon H \ hookrightarrow G}$${\ displaystyle H}$${\ displaystyle G}$

Non-trivial examples

• Consider the additive group of integers and the factor group . The mapping (see congruence and residue class ring ) is a group homomorphism. It is surjective and its core consists of the set of all integers divisible by 3. This homomorphism is called the canonical projection .${\ displaystyle (\ mathbb {Z}, +)}$ ${\ displaystyle (\ mathbb {Z} / 3 \ mathbb {Z}, +) = \ {0 + 3 \ mathbb {Z}, 1 + 3 \ mathbb {Z}, 2 + 3 \ mathbb {Z} \} }$${\ displaystyle p \ colon \ mathbb {Z} \ to \ mathbb {Z} / 3 \ mathbb {Z}, \ p (z) = z \, {\ bmod {\,}} 3 = z + 3 \ mathbb {Z}}$${\ displaystyle 3 \ mathbb {Z}}$
• The exponential function is a group homomorphism between the additive group of real numbers and the multiplicative group of real numbers other than 0, because . This mapping is injective and its image is the set of positive real numbers.${\ displaystyle (\ mathbb {R}, +)}$ ${\ displaystyle \ mathbb {R}}$${\ displaystyle \ left (\ mathbb {R} ^ {*}, \ cdot \ right)}$${\ displaystyle \ operatorname {exp} (x + y) = \ operatorname {exp} (x) \ cdot \ operatorname {exp} (y)}$
• The complex exponential function is a group homomorphism between the complex numbers with the addition and the complex numbers other than 0 with the multiplication. This homomorphism is surjective and its core is how to e.g. B. can be inferred from Euler's identity .${\ displaystyle \ mathbb {C}}$${\ displaystyle \ operatorname {ker} (\ operatorname {exp}) = \ left \ {2 \ pi ki \ colon k \ in \ mathbb {Z} \ right \}}$
• The mapping that assigns its determinant to every invertible matrix is a homomorphism${\ displaystyle n \ times n}$${\ displaystyle GL (n, \ mathbb {R}) \ to (\ mathbb {R}, \, \ cdot)}$
• The mapping that assigns its sign to each permutation is a homomorphism${\ displaystyle S_ {n} \ to (\ {\ pm 1 \}, \, \ cdot)}$

Concatenation of group homomorphisms

If and are two group homomorphisms, then their composition is also a group homomorphism. ${\ displaystyle h \ colon G \ to H}$${\ displaystyle k \ colon H \ to K}$ ${\ displaystyle k \ circ h \ colon G \ to K}$

The class of all groups forms a category with the group homomorphisms .

Mono-, epi-, iso-, endo-, automorphism

A homomorphism is called ${\ displaystyle f \ colon G \ to H}$

If a group isomorphism, then its inverse function is also a group isomorphism, the groups and are then called isomorphic to one another : They only differ in the designation of their elements and agree for almost all purposes. ${\ displaystyle h \ colon G \ to H}$${\ displaystyle G}$${\ displaystyle H}$

If a group homomorphism of a group is in itself, then it is called a group endomorphism. If it is also bijective, then it is called group automorphism. The set of all group endomorphisms of forms a monoid with the composition . The set of all group automorphisms of a group forms a group with the composition, the automorphism group of . ${\ displaystyle h \ colon G \ to G}$${\ displaystyle G}$${\ displaystyle G}$${\ displaystyle \ operatorname {Aut} (G)}$${\ displaystyle G}$

The automorphism group of contains only two elements: the identity (1) and the multiplication by −1; it is isomorphic to the cyclic group . ${\ displaystyle (\ mathbb {Z}, +)}$ ${\ displaystyle C_ {2}}$

In the group of , every linear map with is an automorphism. ${\ displaystyle (\ mathbb {Q}, +)}$ ${\ displaystyle f \ left (x \ right) = m \ cdot x}$${\ displaystyle m \ in \ mathbb {Q} \ setminus \ {0 \}}$

Homomorphisms between Abelian groups

Are and groups, Abelian , then the amount is all group homomorphisms from to even one (again Abelian) group, namely with the "point-wise addition": ${\ displaystyle G}$${\ displaystyle H}$${\ displaystyle H}$ ${\ displaystyle Hom (G, H)}$${\ displaystyle G}$${\ displaystyle H}$

${\ displaystyle \ left (h + k \ right) \ left (x \ right) \ colon = h \ left (x \ right) + k \ left (x \ right)}$for everyone .${\ displaystyle x \ in G}$

The commutativity of is required so that there is a group homomorphism again. ${\ displaystyle H}$${\ displaystyle h + k}$

The set of endomorphisms of an Abelian group forms a group with the addition, which is referred to as. ${\ displaystyle G}$${\ displaystyle End (G)}$

The addition of homomorphisms is compatible with the composition in the following sense: If , then applies ${\ displaystyle f \ in Hom (K, G), h, k \ in Hom (G, H), g \ in Hom (H, L)}$

${\ displaystyle \ left (h + k \ right) \ circ f = \ left (h \ circ f \ right) + \ left (k \ circ f \ right)}$and .${\ displaystyle g \ circ \ left (h + k \ right) = \ left (g \ circ h \ right) + \ left (g \ circ k \ right)}$

This shows that the endomorphism group of an Abelian group even forms a ring , the endomorphism ring of . ${\ displaystyle End (G)}$${\ displaystyle G}$

For example, the ring of endomorphisms of the Klein group of four is isomorphic to the ring of the 2 × 2 matrices over the remainder class field . ${\ displaystyle \ mathbb {Z} / 2 \ mathbb {Z}}$