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Definition

Group homomorphisms

A group is a set G {\displaystyle G} with a binary operation ⋅ {\displaystyle \cdot } satisfying the following three properties for all a , b , c ∈ G {\displaystyle a,b,c\in G} :¹⁰

1. Associative: ( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c ) {\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)}
2. Identity: There is an e ∈ G {\displaystyle e\in G} such that e ⋅ a = a ⋅ e = a {\displaystyle e\cdot a=a\cdot e=a}
3. Inverses: There is an a ′ ∈ G {\displaystyle a'\in G} for each a ∈ G {\displaystyle a\in G} such that a ⋅ a ′ = a ′ ⋅ a = e {\displaystyle a\cdot a'=a'\cdot a=e}

A group is also called abelian if it also satisfies a ⋅ b = b ⋅ a {\displaystyle a\cdot b=b\cdot a} .¹¹

Let G {\displaystyle G} and H {\displaystyle H} be groups. A group homomorphism from G {\displaystyle G} to H {\displaystyle H} is a function f : G → H {\displaystyle f:G\to H} such that f ( a b ) = f ( a ) f ( b ) {\displaystyle f(ab)=f(a)f(b)} for all a , b ∈ G {\displaystyle a,b\in G} .¹² Letting e H {\displaystyle e_{H}} is the identity element of H {\displaystyle H} , then the kernel of f {\displaystyle f} is the preimage of the singleton set { e H } {\displaystyle \{e_{H}\}} ; that is, the subset of G {\displaystyle G} consisting of all those elements of G {\displaystyle G} that are mapped by f {\displaystyle f} to the element e H {\displaystyle e_{H}} .¹³¹⁴

The kernel is usually denoted ker ⁡ f {\displaystyle \ker {f}} (or a variation).¹⁵ In symbols:

ker ⁡ f = { g ∈ G : f ( g ) = e H } . {\displaystyle \ker f=\{g\in G:f(g)=e_{H}\}.}

Since a group homomorphism preserves identity elements, the identity element e G {\displaystyle e_{G}} of G {\displaystyle G} must belong to the kernel.¹⁶ The homomorphism f {\displaystyle f} is injective if and only if its kernel is only the singleton set { e G } {\displaystyle \{e_{G}\}} .¹⁷

ker ⁡ f {\displaystyle \ker {f}} is a subgroup of G {\displaystyle G} and further it is a normal subgroup. Thus, there is a corresponding quotient group G / ker ⁡ f {\displaystyle G/\ker {f}} . This is isomorphic to f ( G ) {\displaystyle f(G)} , the image of G {\displaystyle G} under f {\displaystyle f} (which is a subgroup of H {\displaystyle H} also), by the first isomorphism theorem for groups.¹⁸

Ring homomorphisms

A ring with identity (or unity) is a set R {\displaystyle R} with two binary operations + {\displaystyle +} and ⋅ {\displaystyle \cdot } satisfying:¹⁹²⁰

1. R {\displaystyle R} with + {\displaystyle +} is an abelian group with identity 0 {\displaystyle 0} .
2. Multiplication ⋅ {\displaystyle \cdot } is associative.
3. Distributive: a ⋅ ( b + c ) = a ⋅ b + a ⋅ c {\displaystyle a\cdot (b+c)=a\cdot b+a\cdot c} and ( a + b ) ⋅ c = a ⋅ c + b ⋅ c {\displaystyle (a+b)\cdot c=a\cdot c+b\cdot c} for all a , b , c ∈ R {\displaystyle a,b,c\in R}
4. Multiplication ⋅ {\displaystyle \cdot } has an identity element 1 {\displaystyle 1} .²¹

A ring is commutative if the multiplication is commutative, and such a ring is a field when every 0 ≠ a ∈ R {\displaystyle 0\neq a\in R} has a multiplicative inverse, that is, some b ∈ R {\displaystyle b\in R} where a b = 1 {\displaystyle ab=1} .²² Let R {\displaystyle R} and S {\displaystyle S} be rings. A ring homomorphism from R {\displaystyle R} to S {\displaystyle S} is a function f : R → S {\displaystyle f:R\to S} satisfying for all a , b ∈ R {\displaystyle a,b\in R} :²³

1. f ( a + b ) = f ( a ) + f ( b ) {\displaystyle f(a+b)=f(a)+f(b)}
2. f ( a b ) = f ( a ) f ( b ) {\displaystyle f(ab)=f(a)f(b)}

The kernel of f {\displaystyle f} is the kernel as additive groups.²⁴ It is the preimage of the zero ideal { 0 S } {\displaystyle \{0_{S}\}} , which is, the subset of R {\displaystyle R} consisting of all those elements of R {\displaystyle R} that are mapped by f {\displaystyle f} to the element 0 S {\displaystyle 0_{S}} . The kernel is usually denoted ker ⁡ f {\displaystyle \ker {f}} (or a variation). In symbols:

ker ⁡ f = { r ∈ R : f ( r ) = 0 S } . {\displaystyle \operatorname {ker} f=\{r\in R:f(r)=0_{S}\}.}

Since a ring homomorphism preserves zero elements, the zero element 0 R {\displaystyle 0_{R}} of R {\displaystyle R} must belong to the kernel. The homomorphism f {\displaystyle f} is injective if and only if its kernel is only the singleton set { 0 R } {\displaystyle \{0_{R}\}} . This is always the case if R {\displaystyle R} is a field, and S {\displaystyle S} is not the zero ring.²⁵

Since ker ⁡ f {\displaystyle \ker {f}} contains the multiplicative identity only when S {\displaystyle S} is the zero ring, it turns out that the kernel is generally not a subring of R {\displaystyle R} . The kernel is a subrng, and, more precisely, a two-sided ideal of R {\displaystyle R} . Thus, it makes sense to speak of the quotient ring R / ker ⁡ f {\displaystyle R/\ker {f}} . The first isomorphism theorem for rings states that this quotient ring is naturally isomorphic to the image of f {\displaystyle f} (which is a subring of S {\displaystyle S} ).²⁶

Linear maps

Main article: Kernel (linear algebra)

Given a field F {\displaystyle F} , a vector space (over F {\displaystyle F} ) is an abelian group V {\displaystyle V} (with binary operation + {\displaystyle +} and identity 0 {\displaystyle 0} ) with scalar multiplication from F {\displaystyle F} satisfying for all a , b ∈ F {\displaystyle a,b\in F} and α , β ∈ V {\displaystyle \alpha ,\beta \in V} :²⁷

1. a ( b α ) = ( a b ) α {\displaystyle a(b\alpha )=(ab)\alpha }
2. ( a + b ) α = a α + b α {\displaystyle (a+b)\alpha =a\alpha +b\alpha }
3. a ( α + β ) = a α + a β {\displaystyle a(\alpha +\beta )=a\alpha +a\beta }
4. 1 α = α {\displaystyle 1\alpha =\alpha }

Let V {\displaystyle V} and W {\displaystyle W} be vector spaces over the field F {\displaystyle F} . A linear map (or linear transformation) from V {\displaystyle V} to W {\displaystyle W} is a function T : V → W {\displaystyle T:V\to W} satisfying for all α , β ∈ V {\displaystyle \alpha ,\beta \in V} and a ∈ F {\displaystyle a\in F} :²⁸

1. T ( α + β ) = T ( α ) + T ( β ) {\displaystyle T(\alpha +\beta )=T(\alpha )+T(\beta )}
2. T ( a α ) = a T ( α ) {\displaystyle T(a\alpha )=aT(\alpha )}

If 0 W {\displaystyle 0_{W}} is the zero vector of W {\displaystyle W} , then the kernel of T {\displaystyle T} (or null space²⁹) is the preimage of the zero subspace { 0 W } {\displaystyle \{0_{W}\}} ; that is, the subset of V {\displaystyle V} consisting of all those elements of V {\displaystyle V} that are mapped by T {\displaystyle T} to the element 0 W {\displaystyle 0_{W}} . The kernel is denoted as ker ⁡ T {\displaystyle \ker {T}} , or some variation thereof, and is symbolically defined as:

ker ⁡ T = { v ∈ V : T ( v ) = 0 W } . {\displaystyle \ker T=\{\mathbf {v} \in V:T(\mathbf {v} )=\mathbf {0} _{W}\}.}

Since a linear map preserves zero vectors, the zero vector 0 V {\displaystyle 0_{V}} of V {\displaystyle V} must belong to the kernel. The transformation T {\displaystyle T} is injective if and only if its kernel is reduced to the zero subspace.³⁰

The kernel ker ⁡ T {\displaystyle \ker {T}} is always a linear subspace of V {\displaystyle V} .³¹ Thus, it makes sense to speak of the quotient space V / ker ⁡ T {\displaystyle V/\ker {T}} . The first isomorphism theorem for vector spaces states that this quotient space is naturally isomorphic to the image of T {\displaystyle T} (which is a subspace of W {\displaystyle W} ). As a consequence, the dimension of V {\displaystyle V} equals the dimension of the kernel plus the dimension of the image.³²

Module homomorphisms

Let R {\displaystyle R} be a ring. A modules over R {\displaystyle R} is defined exactly like a vector space over a field, using the same axioms, expect the field is replaced with a ring. In fact, a module over a field is exactly the same as a vector space over a field.³³ Let M {\displaystyle M} and N {\displaystyle N} be R {\displaystyle R} -modules. A module homomorphism from M {\displaystyle M} to N {\displaystyle N} is also a function φ : M → N {\displaystyle \varphi :M\to N} satisfying the same analogous properties of a linear map. The kernel of φ {\displaystyle \varphi } is defined to be:³⁴

ker ⁡ φ = { m ∈ M   |   φ ( m ) = 0 } {\displaystyle \ker \varphi =\{m\in M\ |\ \varphi (m)=0\}}

Every kernel is a submodule of the domain module, which means they always contain 0, the additive identity of the module. Kernels of abelian groups can be considered a particular kind of module kernel when the underlying ring is the integers.³⁵

Examples

Group homomorphisms

Let G {\displaystyle G} be the cyclic group on 6 elements { 0 , 1 , 2 , 3 , 4 , 5 , } {\displaystyle \{0,1,2,3,4,5,\}} with modular addition, H {\displaystyle H} be the cyclic on 2 elements { 0 , 1 } {\displaystyle \{0,1\}} with modular addition, and f {\displaystyle f} the homomorphism that maps each element g ∈ G {\displaystyle g\in G} to the element g {\displaystyle g} modulo 2 in H {\displaystyle H} . Then ker ⁡ f = { 0 , 2 , 4 } {\displaystyle \ker f=\{0,2,4\}} , since all these elements are mapped to 0 ∈ H {\displaystyle 0\in H} . The quotient group G / ker ⁡ f {\displaystyle G/\ker {f}} has two elements: { 0 , 2 , 4 } {\displaystyle \{0,2,4\}} and { 1 , 3 , 5 } {\displaystyle \{1,3,5\}} , and is isomorphic to H {\displaystyle H} .³⁶

Given a isomorphism φ : G → H {\displaystyle \varphi :G\to H} , one has ker ⁡ φ = 1 {\displaystyle \ker \varphi =1} .³⁷ On the other hand, if this mapping is merely a homomorphism where H is the trivial group, then φ ( g ) = 1 {\displaystyle \varphi (g)=1} for all g ∈ G {\displaystyle g\in G} , so thus ker ⁡ φ = G {\displaystyle \ker \varphi =G} .³⁸

Let φ : R 2 → R {\displaystyle \varphi :\mathbb {R} ^{2}\to \mathbb {R} } be the map defined as φ ( ( x , y ) ) = x {\displaystyle \varphi ((x,y))=x} . Then this is a homomorphism with the kernel consisting precisely the points of the form ( 0 , y ) {\displaystyle (0,y)} . This mapping is considered the "projection onto the x-axis."³⁹ A similar phenomenon occurs with the mapping f : ( R × ) 2 → R × {\displaystyle f:(\mathbb {R} ^{\times })^{2}\to \mathbb {R} ^{\times }} defined as f ( a , b ) = b {\displaystyle f(a,b)=b} , where the kernel is the points of the form ( a , 1 ) {\displaystyle (a,1)} ⁴⁰

For a non-abelian example, let Q 8 {\displaystyle Q_{8}} denote the Quaternion group, and V 4 {\displaystyle V_{4}} the Klein 4-group. Define a mapping φ : Q 8 → V 4 {\displaystyle \varphi :Q_{8}\to V_{4}} to be:⁴¹

φ ( ± 1 ) = 1 {\displaystyle \varphi (\pm 1)=1} φ ( ± i ) = a {\displaystyle \varphi (\pm i)=a} φ ( ± j ) = b {\displaystyle \varphi (\pm j)=b} φ ( ± k ) = c {\displaystyle \varphi (\pm k)=c}

Then this mapping is a homomorphism where ker ⁡ φ = { ± 1 } {\displaystyle \ker \varphi =\{\pm 1\}} .⁴²

Let S 1 {\displaystyle S^{1}} denote the circle group, consisting of all complex numbers with absolute value (or modulus) of 1 {\displaystyle 1} , with the group operation being multiplication.⁴³ Then the function f : R → S 1 {\displaystyle f:\mathbb {R} \to S^{1}} sending x ↦ e 2 π i x = cos ⁡ ( 2 π x ) + i sin ⁡ ( 2 π x ) {\displaystyle x\mapsto e^{2\pi ix}=\cos(2\pi x)+i\sin(2\pi x)} is a homomorphism with the integers being the kernel. The first isomorphism theorem then implies that R / Z ≅ S 1 {\displaystyle \mathbb {R} /\mathbb {Z} \cong S^{1}} .⁴⁴

The symmetric group on n {\displaystyle n} elements, S n {\displaystyle S_{n}} , has a surjective homomorphism ϵ : S n → Z 2 {\displaystyle \epsilon :S_{n}\to \mathbb {Z} _{2}} that takes each permutation to the parity of the number of transpositions whose product is that permutation. The alternating group A n = ker ⁡ ϵ {\displaystyle A_{n}=\ker \epsilon } is the kernel of this homomorphism, consisting of the even permutations. The alternating group is a non-abelian simple group for n ≥ 5 {\displaystyle n\geq 5} .⁴⁵

The determinant of n × n {\displaystyle n\times n} invertible matrices of the real numbers R {\displaystyle \mathbb {R} } , whose set is denoted G L ( n , R ) {\displaystyle GL(n,\mathbb {R} )} and called the general linear group of n × n {\displaystyle n\times n} matrices of R {\displaystyle \mathbb {R} } , is a homomorphism onto the multiplication group R × {\displaystyle \mathbb {R} ^{\times }} (consisting of all non-zero real numbers), and the kernel of the determinant is called the special linear group S L ( n , R ) {\displaystyle SL(n,\mathbb {R} )} of n × n {\displaystyle n\times n} matrices of R {\displaystyle \mathbb {R} } . These are the matrices whose determinant is precisely 1 {\displaystyle 1} .⁴⁶

Given a group G {\displaystyle G} and an element, the mapping x ↦ g x g − 1 {\displaystyle x\mapsto gxg^{-1}} is an automorphism - an isomorphism whose domain and image are the same group. This gives a homomorphism from G {\displaystyle G} to its automorphism group Aut ( G ) {\displaystyle {\text{Aut}}(G)} , mapping each g {\displaystyle g} to its respective inner automorphism as described, and the kernel of this homomorphism is the center Z ( G ) {\displaystyle Z(G)} of G {\displaystyle G} , consisting of g ∈ G {\displaystyle g\in G} where for every x ∈ G {\displaystyle x\in G} , we have g x g − 1 = x {\displaystyle gxg^{-1}=x} , or equivalently g x = x g {\displaystyle gx=xg} . More generally, for every normal subgroup H {\displaystyle H} of G {\displaystyle G} (i.e. groups closed under conjugation), this conjugation map is also an automorphism on H {\displaystyle H} , giving another homomorphism G {\displaystyle G} to Aut ( H ) {\displaystyle {\text{Aut}}(H)} , with the kernel being the centralizer C G ( H ) {\displaystyle C_{G}(H)} of H {\displaystyle H} in G {\displaystyle G} , being the set of g ∈ G {\displaystyle g\in G} where for every h ∈ H {\displaystyle h\in H} , we have g h g − 1 = h {\displaystyle ghg^{-1}=h} .⁴⁷

Ring homomorphisms

Consider the mapping φ : Z → Z / 2 Z {\displaystyle \varphi :\mathbb {Z} \to \mathbb {Z} /2\mathbb {Z} } where the later ring is the integers modulo 2 and the map sends each number to its parity; 0 for even numbers, and 1 for odd numbers. This mapping turns out to be a homomorphism, and since the additive identity of the later ring is 0, the kernel is precisely the even numbers.⁴⁸

Let φ : Q [ x ] → Q {\displaystyle \varphi :\mathbb {Q} [x]\to \mathbb {Q} } be defined as φ ( p ( x ) ) = p ( 0 ) {\displaystyle \varphi (p(x))=p(0)} . This mapping, which happens to be a homomorphism, sends each polynomial to its constant term. It maps a polynomial to zero if and only if said polynomial's constant term is 0.⁴⁹ Polynomials with real coefficients can receive a similar homomorphism, with its kernel being the polynomials with constant term 0.⁵⁰

Linear maps

Let φ : C 3 → C {\displaystyle \varphi :\mathbb {C} ^{3}\to \mathbb {C} } be defined as φ ( x , y , z ) = x + 2 y + 3 z {\displaystyle \varphi (x,y,z)=x+2y+3z} , then the kernel of φ {\displaystyle \varphi } (that is, the null space) will be the set of points ( x , y , z ) ∈ C 3 {\displaystyle (x,y,z)\in \mathbb {C} ^{3}} such that x + 2 y + 3 z = 0 {\displaystyle x+2y+3z=0} , and this set is a subspace of C 3 {\displaystyle \mathbb {C} ^{3}} (the same is true for every kernel of a linear map).⁵¹

If D {\displaystyle D} represents the derivative operator on real polynomials, then the kernel of D {\displaystyle D} will consist of the polynomials with deterivative equal to 0, that is the constant functions.⁵²

Consider the mapping ( T p ) ( x ) = x 2 p ( x ) {\displaystyle (Tp)(x)=x^{2}p(x)} , where p {\displaystyle p} is a polynomial with real coefficients. Then T {\displaystyle T} is a linear map whose kernel is precisely 0, since 0 is the only polynomial to satisfy x 2 p ( x ) = 0 {\displaystyle x^{2}p(x)=0} for all x ∈ R {\displaystyle x\in \mathbb {R} } .⁵³

Quotient algebras

The kernel of a homomorphism can be used to define a quotient algebra. Let G {\displaystyle G} and H {\displaystyle H} be groups, φ : G → H {\displaystyle \varphi :G\to H} be a group homomorphism, and denote K = ker ⁡ φ {\displaystyle K=\ker \varphi } . Put G / K {\displaystyle G/K} to be the set of fibers of the homomorphism φ {\displaystyle \varphi } , where a fiber is the set of points of the domain mapping to a single point in the range.⁵⁴ Let X a ∈ G / K {\displaystyle X_{a}\in G/K} denotes the fiber of the element a ∈ H {\displaystyle a\in H} , then a group operation on the set of fibers can be endowed by X a X b = X a b {\displaystyle X_{a}X_{b}=X_{ab}} , and G / K {\displaystyle G/K} is called the quotient group (or factor group), to be read as "G modulo K" or "G mod K".⁵⁵ The terminology arises from the fact that the kernel represents the fiber of the identity element of the range, H {\displaystyle H} , and that the remaining elements are simply "translates" of the kernel, so the quotient group is obtained by "dividing out" the kernel.⁵⁶

The fibers can also be described by looking at the domain relative to the kernel; given X ∈ G / K {\displaystyle X\in G/K} and any element u ∈ X {\displaystyle u\in X} , then X = u K = K u {\displaystyle X=uK=Ku} where:⁵⁷

u K = { u k   |   k ∈ K } {\displaystyle uK=\{uk\ |\ k\in K\}} K u = { k u   |   k ∈ K } {\displaystyle Ku=\{ku\ |\ k\in K\}}

these sets are called the left and right cosets respectively, and can be defined in general for any arbitrary subgroup of G {\displaystyle G} .⁵⁸⁵⁹⁶⁰ The group operation can then be defined as u K ∘ v K = ( u k ) K {\displaystyle uK\circ vK=(uk)K} , which is well-defined regardless of the choice of representatives of the fibers.⁶¹⁶²

According to the first isomorphism theorem, there is an isomorphism μ : G / K → φ ( G ) {\displaystyle \mu :G/K\to \varphi (G)} , where the later group is the image of the homomorphism φ {\displaystyle \varphi } , and the isomorphism is defined as μ ( u K ) = φ ( u ) {\displaystyle \mu (uK)=\varphi (u)} , and such map is also well-defined.⁶³⁶⁴

For rings, modules, and vector spaces, one can define the respective quotient algebras via the underlying additive group structure, with cosets represented as x + K {\displaystyle x+K} . Ring multiplication can be defined on the quotient algebra as ( x + K ) ( y + K ) = x y + K {\displaystyle (x+K)(y+K)=xy+K} , and is well-defined.⁶⁵ For a ring R {\displaystyle R} (possibly a field when describing vector spaces) and a module homomorphism φ : M → N {\displaystyle \varphi :M\to N} with kernel K = ker ⁡ φ {\displaystyle K=\ker \varphi } , one can define scalar multiplication on G / K {\displaystyle G/K} by r ( x + K ) = r x + K {\displaystyle r(x+K)=rx+K} for r ∈ R {\displaystyle r\in R} and x ∈ M {\displaystyle x\in M} , which will also be well-defined.⁶⁶

Kernel structures

The structure of kernels allows for the building of quotient algebras from structures satisfying the properties of kernels. Any subgroup N {\displaystyle N} of a group G {\displaystyle G} can construct a quotient G / N {\displaystyle G/N} by the set of all cosets of N {\displaystyle N} in G {\displaystyle G} .⁶⁷ The natural way to turn this into a group, similar to the treatment for the quotient by a kernel, is to define an operation on (left) cosets by u N ⋅ v N = ( u v ) N {\displaystyle uN\cdot vN=(uv)N} , however this operation is well defined if and only if the subgroup N {\displaystyle N} is closed under conjugation under G {\displaystyle G} , that is, if g ∈ G {\displaystyle g\in G} and n ∈ N {\displaystyle n\in N} , then g n g − 1 ∈ N {\displaystyle gng^{-1}\in N} . Furthermore, the operation being well defined is sufficient for the quotient to be a group.⁶⁸ Subgroups satisfying this property are called normal subgroups.⁶⁹ Every kernel of a group is a normal subgroup, and for a given normal subgroup N {\displaystyle N} of a group G {\displaystyle G} , the natural projection π : G → G / N {\displaystyle \pi :G\to G/N} defined as π ( g ) = g N {\displaystyle \pi (g)=gN} is a homomorphism with ker ⁡ π = N {\displaystyle \ker \pi =N} , so the normal subgroups are precisely the subgroups which are kernels.⁷⁰ The closure under conjugation, however, gives a criterion for when a subgroup is a kernel for some homomorphism.⁷¹

For a ring R {\displaystyle R} , treating it as a group, one can take a quotient group via an arbitrary subgroup I {\displaystyle I} of the ring, which will be normal due to the ring's additive group being abelian. To define multiplication on R / I {\displaystyle R/I} , the multiplication of cosets, defined as ( r + I ) ( s + I ) = r s + I {\displaystyle (r+I)(s+I)=rs+I} needs to be well-defined. Taking representatives r + α {\displaystyle r+\alpha } and s + β {\displaystyle s+\beta } of r + I {\displaystyle r+I} and s + I {\displaystyle s+I} respectively, for r , s ∈ R {\displaystyle r,s\in R} and α , β ∈ I {\displaystyle \alpha ,\beta \in I} , yields:⁷²

( r + α ) ( s + β ) + I = r s + I {\displaystyle (r+\alpha )(s+\beta )+I=rs+I}

Setting r = s = 0 {\displaystyle r=s=0} implies that I {\displaystyle I} is closed under multiplication, while setting α = s = 0 {\displaystyle \alpha =s=0} shows that r β ∈ I {\displaystyle r\beta \in I} , that is, I {\displaystyle I} is closed under arbitrary multiplication by elements on the left. Similarly, taking r = β = 0 {\displaystyle r=\beta =0} implies that I {\displaystyle I} is also closed under multiplication by arbitrary elements on the right.⁷³ Any subgroup of R {\displaystyle R} that is closed under multiplication by any element of the ring is called an ideal.⁷⁴ Analogously to normal subgroups, the ideals of a ring are precisely the kernels of homomorphisms.⁷⁵

Exact sequence

Main article: Exact sequence

Kernels are used to define exact sequences of homomorphisms for groups and modules. Given modules A {\displaystyle A} , B {\displaystyle B} , and C {\displaystyle C} , a pair of homomorphisms ψ : A → B , φ : B → C {\displaystyle \psi :A\to B,\varphi :B\to C} is said to be exact if image  ψ = ker ⁡ φ {\displaystyle {\text{image }}\psi =\ker \varphi } . An exact sequence is then a sequence of modules and homomorphisms ⋯ → X n − 1 → X n → X n + 1 → ⋯ {\displaystyle \cdots \to X_{n-1}\to X_{n}\to X_{n+1}\to \cdots } where each adjacent pair of modules and homomorphisms is exact.⁷⁶

Universal algebra

Kernels can be generalized in universal algebra for homomorphisms between any two algebraic structures. An operation on a set A {\displaystyle A} is a function of the form Q : A n → A {\displaystyle Q:A^{n}\to A} , where n {\displaystyle n} is the arity (or rank) of the operation. An n {\displaystyle n} -ary operation takes an ordered list of n {\displaystyle n} elements from A {\displaystyle A} and maps them to a single element in A {\displaystyle A} . An algebraic structure is a tuple ⟨ A , F ⟩ {\displaystyle \langle A,F\rangle } where A {\displaystyle A} is the underlying set of the algebra, and F {\displaystyle F} is an indexed set of operations Q ∈ F {\displaystyle Q\in F} on A {\displaystyle A} , with their interpretation denoted Q A {\displaystyle Q^{A}} . The set indexing F {\displaystyle F} is the language, which also maps each operation symbol to their fixed arity (called the rank function). Two algebraic structures are similar when they share the same language, including their rank function.⁷⁷⁷⁸

Let A {\displaystyle A} and B {\displaystyle B} be algebraic structures of a similar type F {\displaystyle F} . A homomorphism is a function f : A → B {\displaystyle f:A\to B} that respects the interpretation of each Q ∈ F {\displaystyle Q\in F} , that is, taking Q {\displaystyle Q} to be an n {\displaystyle n} -ary operation, and a i ∈ A {\displaystyle a_{i}\in A} for 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} : ⁷⁹⁸⁰

f ( Q A ( a 1 , … a n ) ) = Q B ( f ( a 1 ) , … f ( a n ) ) {\displaystyle f(Q^{A}(a_{1},\ldots a_{n}))=Q^{B}(f(a_{1}),\ldots f(a_{n}))}

The kernel of f {\displaystyle f} , denoted ker ⁡ f {\displaystyle \ker {f}} , is the subset of the direct product A × A {\displaystyle A\times A} consisting of all ordered pairs of elements of A {\displaystyle A} whose components are both mapped by f {\displaystyle f} to the same element in B {\displaystyle B} . In symbols:⁸¹⁸²

ker ⁡ f = { ( a , b ) ∈ A × A : f ( a ) = f ( b ) } . {\displaystyle \ker f=\left\{\left(a,b\right)\in A\times A:f(a)=f\left(b\right)\right\}{\mbox{.}}}

The homomorphism f {\displaystyle f} is injective if and only if its kernel is the diagonal set { ( a , a )   |   a ∈ A } {\displaystyle \{(a,a)\ |\ a\in A\}} , which is always contained inside the kernel.⁸³⁸⁴ ker ⁡ f {\displaystyle \ker {f}} is an equivalence relation on A {\displaystyle A} , and in fact a congruence relation, meaning that for an n-ary operation Q ∈ F {\displaystyle Q\in F} , the relation a i ker ⁡ f   b i {\displaystyle a_{i}\ker {f}\ b_{i}} for 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} implies Q A ( a 1 , … a n ) ker ⁡ f   Q A ( b 1 , … b n ) {\displaystyle Q^{A}(a_{1},\ldots a_{n})\ker {f}\ Q^{A}(b_{1},\ldots b_{n})} . It makes sense to speak of the quotient algebra A / ker ⁡ f {\displaystyle A/\ker {f}} , with the set consisting of the equivalence classes of the kernel, and the well-defined operations defined for an n {\displaystyle n} -ary operation Q ∈ F {\displaystyle Q\in F} as: ⁸⁵

Q A / ker ⁡ f ( a 1 / ker ⁡ f , … a n / ker ⁡ f ) = Q A ( a 1 , … a n ) / ker ⁡ f {\displaystyle Q^{A/\ker {f}}(a_{1}/\ker {f},\ldots a_{n}/\ker {f})=Q^{A}(a_{1},\ldots a_{n})/\ker {f}}

The first isomorphism theorem in universal algebra states that this quotient algebra is naturally isomorphic to the image of f {\displaystyle f} (which is a subalgebra of B {\displaystyle B} ).⁸⁶

See also

• Kernel (linear algebra)
• Kernel (category theory)
• Kernel of a function
• Equalizer (mathematics)
• Zero set

Notes

Sources

• Axler, Sheldon. Linear Algebra Done Right (4th ed.). Springer.
• Burris, Stanley; Sankappanavar, H.P. (2012). A Course in Universal Algebra (Millennium ed.). S. Burris and H.P. Sankappanavar. ISBN 978-0-9880552-0-9.
• Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7.
• Fraleigh, John B.; Katz, Victor (2003). A first course in abstract algebra. World student series (7th ed.). Boston: Addison-Wesley. ISBN 978-0-201-76390-4.
• Hungerford, Thomas W. (2014). Abstract Algebra: an introduction (3rd ed.). Boston, MA: Brooks/Cole, Cengage Learning. ISBN 978-1-111-56962-4.
• McKenzie, Ralph; McNulty, George F.; Taylor, W. (1987). Algebras, lattices, varieties. The Wadsworth & Brooks/Cole mathematics series. Monterey, Calif: Wadsworth & Brooks/Cole Advanced Books & Software. ISBN 978-0-534-07651-1.
• Rotman, Joseph J. (2002). Advanced modern algebra. Upper Saddle River, NJ: Prentice Hall. ISBN 0130878685.

References

1. McKenzie, McNulty & Taylor 1987, pp. 27–29 - McKenzie, Ralph; McNulty, George F.; Taylor, W. (1987). Algebras, lattices, varieties. The Wadsworth & Brooks/Cole mathematics series. Monterey, Calif: Wadsworth & Brooks/Cole Advanced Books & Software. ISBN 978-0-534-07651-1. ↩

2. Dummit & Foote 2004, p. 75 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

3. Dummit & Foote 2004, p. 240 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

4. Dummit & Foote 2004, p. 97 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

5. Dummit & Foote 2004, p. 82 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

6. Dummit & Foote 2004, pp. 239–247 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

7. McKenzie, McNulty & Taylor 1987, pp. 27–29 - McKenzie, Ralph; McNulty, George F.; Taylor, W. (1987). Algebras, lattices, varieties. The Wadsworth & Brooks/Cole mathematics series. Monterey, Calif: Wadsworth & Brooks/Cole Advanced Books & Software. ISBN 978-0-534-07651-1. ↩

8. McKenzie, McNulty & Taylor 1987, pp. 27–29 - McKenzie, Ralph; McNulty, George F.; Taylor, W. (1987). Algebras, lattices, varieties. The Wadsworth & Brooks/Cole mathematics series. Monterey, Calif: Wadsworth & Brooks/Cole Advanced Books & Software. ISBN 978-0-534-07651-1. ↩

9. Dummit & Foote 2004, p. 97 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

10. Fraleigh & Katz 2003, pp. 23, 37–39 - Fraleigh, John B.; Katz, Victor (2003). A first course in abstract algebra. World student series (7th ed.). Boston: Addison-Wesley. ISBN 978-0-201-76390-4. ↩

11. Fraleigh & Katz 2003, pp. 23, 37–39 - Fraleigh, John B.; Katz, Victor (2003). A first course in abstract algebra. World student series (7th ed.). Boston: Addison-Wesley. ISBN 978-0-201-76390-4. ↩

12. Fraleigh & Katz 2003, p. 125 - Fraleigh, John B.; Katz, Victor (2003). A first course in abstract algebra. World student series (7th ed.). Boston: Addison-Wesley. ISBN 978-0-201-76390-4. ↩

13. Dummit & Foote 2004, p. 75 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

14. Hungerford 2014, p. 263 - Hungerford, Thomas W. (2014). Abstract Algebra: an introduction (3rd ed.). Boston, MA: Brooks/Cole, Cengage Learning. ISBN 978-1-111-56962-4. ↩

15. Dummit & Foote 2004, p. 75 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

16. Dummit & Foote 2004, p. 75 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

17. Hungerford 2014, p. 264 - Hungerford, Thomas W. (2014). Abstract Algebra: an introduction (3rd ed.). Boston, MA: Brooks/Cole, Cengage Learning. ISBN 978-1-111-56962-4. ↩

18. Dummit & Foote 2004, p. 97 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

19. Fraleigh & Katz 2003, pp. 167, 172 - Fraleigh, John B.; Katz, Victor (2003). A first course in abstract algebra. World student series (7th ed.). Boston: Addison-Wesley. ISBN 978-0-201-76390-4. ↩

20. Dummit & Foote 2004, pp. 223–224 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

21. Some sources[11][12] do not include the multiplicative identity 1 {\displaystyle 1} in the definition of a ring. ↩

22. Dummit & Foote 2004, pp. 223–224 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

23. Fraleigh & Katz 2003, p. 171 - Fraleigh, John B.; Katz, Victor (2003). A first course in abstract algebra. World student series (7th ed.). Boston: Addison-Wesley. ISBN 978-0-201-76390-4. ↩

24. Fraleigh & Katz 2003, p. 238 - Fraleigh, John B.; Katz, Victor (2003). A first course in abstract algebra. World student series (7th ed.). Boston: Addison-Wesley. ISBN 978-0-201-76390-4. ↩

25. Dummit & Foote 2004, pp. 239–247 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

26. Dummit & Foote 2004, pp. 239–247 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

27. Fraleigh & Katz 2003, pp. 274–275 - Fraleigh, John B.; Katz, Victor (2003). A first course in abstract algebra. World student series (7th ed.). Boston: Addison-Wesley. ISBN 978-0-201-76390-4. ↩

28. Fraleigh & Katz 2003, p. 282 - Fraleigh, John B.; Katz, Victor (2003). A first course in abstract algebra. World student series (7th ed.). Boston: Addison-Wesley. ISBN 978-0-201-76390-4. ↩

29. Axler, p. 59 - Axler, Sheldon. Linear Algebra Done Right (4th ed.). Springer. ↩

30. Axler, p. 60 - Axler, Sheldon. Linear Algebra Done Right (4th ed.). Springer. ↩

31. Dummit & Foote 2004, p. 413 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

32. Dummit & Foote 2004, p. 413 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

33. Dummit & Foote 2004, p. 337 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

34. Dummit & Foote 2004, pp. 345–346 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

35. Dummit & Foote 2004, pp. 345–346 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

36. Dummit & Foote 2004, pp. 78–80 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

37. Dummit & Foote 2004, pp. 78–80 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

38. Dummit & Foote 2004, pp. 78–80 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

39. Dummit & Foote 2004, pp. 78–80 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

40. Hungerford 2014, p. 263 - Hungerford, Thomas W. (2014). Abstract Algebra: an introduction (3rd ed.). Boston, MA: Brooks/Cole, Cengage Learning. ISBN 978-1-111-56962-4. ↩

41. Dummit & Foote 2004, pp. 78–80 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

42. Dummit & Foote 2004, pp. 78–80 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

43. Rotman 2003, p. 53 harvnb error: no target: CITEREFRotman2003 (help) ↩

44. Rotman 2003, pp. 86–87 harvnb error: no target: CITEREFRotman2003 (help) ↩

45. Dummit & Foote 2004, pp. 106–111 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

46. Rotman 2003, p. 76 harvnb error: no target: CITEREFRotman2003 (help) ↩

47. Dummit & Foote 2004, pp. 133–134 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

48. Dummit & Foote 2004, p. 240 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

49. Dummit & Foote 2004, p. 240 - Dummit, David Steven; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 978-0-471-43334-7. ↩

50. Hungerford 2014, p. 155 - Hungerford, Thomas W. (2014). Abstract Algebra: an introduction (3rd ed.). Boston, MA: Brooks/Cole, Cengage Learning. ISBN 978-1-111-56962-4. ↩

51. Axler, p. 59 - Axler, Sheldon. Linear Algebra Done Right (4th ed.). Springer. ↩

52. Axler, p. 59 - Axler, Sheldon. Linear Algebra Done Right (4th ed.). Springer. ↩

53. Axler, p. 59 - Axler, Sheldon. Linear Algebra Done Right (4th ed.). Springer. ↩

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59. Hungerford 2014, pp. 237–239 - Hungerford, Thomas W. (2014). Abstract Algebra: an introduction (3rd ed.). Boston, MA: Brooks/Cole, Cengage Learning. ISBN 978-1-111-56962-4. ↩

60. Fraleigh & Katz 2003, p. 97 - Fraleigh, John B.; Katz, Victor (2003). A first course in abstract algebra. World student series (7th ed.). Boston: Addison-Wesley. ISBN 978-0-201-76390-4. ↩

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62. Fraleigh & Katz 2003, p. 138 - Fraleigh, John B.; Katz, Victor (2003). A first course in abstract algebra. World student series (7th ed.). Boston: Addison-Wesley. ISBN 978-0-201-76390-4. ↩

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64. Fraleigh & Katz 2003, p. 307 - Fraleigh, John B.; Katz, Victor (2003). A first course in abstract algebra. World student series (7th ed.). Boston: Addison-Wesley. ISBN 978-0-201-76390-4. ↩

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77. Burris & Sankappanavar 2012, p. 23 - Burris, Stanley; Sankappanavar, H.P. (2012). A Course in Universal Algebra (Millennium ed.). S. Burris and H.P. Sankappanavar. ISBN 978-0-9880552-0-9. ↩

78. McKenzie, McNulty & Taylor 1987, pp. 11–13 - McKenzie, Ralph; McNulty, George F.; Taylor, W. (1987). Algebras, lattices, varieties. The Wadsworth & Brooks/Cole mathematics series. Monterey, Calif: Wadsworth & Brooks/Cole Advanced Books & Software. ISBN 978-0-534-07651-1. ↩

79. Burris & Sankappanavar 2012, p. 28 - Burris, Stanley; Sankappanavar, H.P. (2012). A Course in Universal Algebra (Millennium ed.). S. Burris and H.P. Sankappanavar. ISBN 978-0-9880552-0-9. ↩

80. McKenzie, McNulty & Taylor 1987, p. 20 - McKenzie, Ralph; McNulty, George F.; Taylor, W. (1987). Algebras, lattices, varieties. The Wadsworth & Brooks/Cole mathematics series. Monterey, Calif: Wadsworth & Brooks/Cole Advanced Books & Software. ISBN 978-0-534-07651-1. ↩

81. Burris & Sankappanavar 2012, p. 44 - Burris, Stanley; Sankappanavar, H.P. (2012). A Course in Universal Algebra (Millennium ed.). S. Burris and H.P. Sankappanavar. ISBN 978-0-9880552-0-9. ↩

82. McKenzie, McNulty & Taylor 1987, pp. 27–29 - McKenzie, Ralph; McNulty, George F.; Taylor, W. (1987). Algebras, lattices, varieties. The Wadsworth & Brooks/Cole mathematics series. Monterey, Calif: Wadsworth & Brooks/Cole Advanced Books & Software. ISBN 978-0-534-07651-1. ↩

83. Burris & Sankappanavar 2012, p. 50 - Burris, Stanley; Sankappanavar, H.P. (2012). A Course in Universal Algebra (Millennium ed.). S. Burris and H.P. Sankappanavar. ISBN 978-0-9880552-0-9. ↩

84. McKenzie, McNulty & Taylor 1987, pp. 27–29 - McKenzie, Ralph; McNulty, George F.; Taylor, W. (1987). Algebras, lattices, varieties. The Wadsworth & Brooks/Cole mathematics series. Monterey, Calif: Wadsworth & Brooks/Cole Advanced Books & Software. ISBN 978-0-534-07651-1. ↩

85. Burris & Sankappanavar 2012, p. 36 - Burris, Stanley; Sankappanavar, H.P. (2012). A Course in Universal Algebra (Millennium ed.). S. Burris and H.P. Sankappanavar. ISBN 978-0-9880552-0-9. ↩

86. Burris & Sankappanavar 2012, pp. 44–46 - Burris, Stanley; Sankappanavar, H.P. (2012). A Course in Universal Algebra (Millennium ed.). S. Burris and H.P. Sankappanavar. ISBN 978-0-9880552-0-9. ↩