• Source: Group homomorphism

In mathematics, given two groups, (G,∗) and (H, ·), a group homomorphism from (G,∗) to (H, ·) is a function h : G → H such that for all u and v in G it holds that




h
(
u
∗
v
)
=
h
(
u
)
⋅
h
(
v
)


{\displaystyle h(u*v)=h(u)\cdot h(v)}


where the group operation on the left side of the equation is that of G and on the right side that of H.
From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H,




h
(

e

G


)
=

e

H




{\displaystyle h(e_{G})=e_{H}}


and it also maps inverses to inverses in the sense that




h

(

u

−
1


)

=
h
(
u

)

−
1


.



{\displaystyle h\left(u^{-1}\right)=h(u)^{-1}.\,}


Hence one can say that h "is compatible with the group structure".
In areas of mathematics where one considers groups endowed with additional structure, a homomorphism sometimes means a map which respects not only the group structure (as above) but also the extra structure. For example, a homomorphism of topological groups is often required to be continuous.




Properties


Let




e

H




{\displaystyle e_{H}}

be the identity element of the (H, ·) group and



u
∈
G


{\displaystyle u\in G}

, then




h
(
u
)
⋅

e

H


=
h
(
u
)
=
h
(
u
∗

e

G


)
=
h
(
u
)
⋅
h
(

e

G


)


{\displaystyle h(u)\cdot e_{H}=h(u)=h(u*e_{G})=h(u)\cdot h(e_{G})}


Now by multiplying for the inverse of



h
(
u
)


{\displaystyle h(u)}

(or applying the cancellation rule) we obtain





e

H


=
h
(

e

G


)


{\displaystyle e_{H}=h(e_{G})}


Similarly,





e

H


=
h
(

e

G


)
=
h
(
u
∗

u

−
1


)
=
h
(
u
)
⋅
h
(

u

−
1


)


{\displaystyle e_{H}=h(e_{G})=h(u*u^{-1})=h(u)\cdot h(u^{-1})}


Therefore for the uniqueness of the inverse:



h
(

u

−
1


)
=
h
(
u

)

−
1




{\displaystyle h(u^{-1})=h(u)^{-1}}

.




Types


Monomorphism
A group homomorphism that is injective (or, one-to-one); i.e., preserves distinctness.
Epimorphism
A group homomorphism that is surjective (or, onto); i.e., reaches every point in the codomain.
Isomorphism
A group homomorphism that is bijective; i.e., injective and surjective. Its inverse is also a group homomorphism. In this case, the groups G and H are called isomorphic; they differ only in the notation of their elements (except of identity element) and are identical for all practical purposes. I.e. we re-label all elements except identity.
Endomorphism
A group homomorphism, h: G → G; the domain and codomain are the same. Also called an endomorphism of G.
Automorphism
A group endomorphism that is bijective, and hence an isomorphism. The set of all automorphisms of a group G, with functional composition as operation, itself forms a group, the automorphism group of G. It is denoted by Aut(G). As an example, the automorphism group of (Z, +) contains only two elements, the identity transformation and multiplication with −1; it is isomorphic to (Z/2Z, +).




Image and kernel



We define the kernel of h to be the set of elements in G which are mapped to the identity in H




ker
⁡
(
h
)
:=

{

u
∈
G
:
h
(
u
)
=

e

H



}

.


{\displaystyle \operatorname {ker} (h):=\left\{u\in G\colon h(u)=e_{H}\right\}.}


and the image of h to be




im
⁡
(
h
)
:=
h
(
G
)
≡

{

h
(
u
)
:
u
∈
G

}

.


{\displaystyle \operatorname {im} (h):=h(G)\equiv \left\{h(u)\colon u\in G\right\}.}


The kernel and image of a homomorphism can be interpreted as measuring how close it is to being an isomorphism. The first isomorphism theorem states that the image of a group homomorphism, h(G) is isomorphic to the quotient group G/ker h.
The kernel of h is a normal subgroup of G. Assume



u
∈
ker
⁡
(
h
)


{\displaystyle u\in \operatorname {ker} (h)}

and show




g

−
1


∘
u
∘
g
∈
ker
⁡
(
h
)


{\displaystyle g^{-1}\circ u\circ g\in \operatorname {ker} (h)}

for arbitrary



u
,
g


{\displaystyle u,g}

:








h

(


g

−
1


∘
u
∘
g

)




=
h
(
g

)

−
1


⋅
h
(
u
)
⋅
h
(
g
)






=
h
(
g

)

−
1


⋅

e

H


⋅
h
(
g
)






=
h
(
g

)

−
1


⋅
h
(
g
)
=

e

H


,






{\displaystyle {\begin{aligned}h\left(g^{-1}\circ u\circ g\right)&=h(g)^{-1}\cdot h(u)\cdot h(g)\\&=h(g)^{-1}\cdot e_{H}\cdot h(g)\\&=h(g)^{-1}\cdot h(g)=e_{H},\end{aligned}}}


The image of h is a subgroup of H.
The homomorphism, h, is a group monomorphism; i.e., h is injective (one-to-one) if and only if ker(h) = {eG}. Injection directly gives that there is a unique element in the kernel, and, conversely, a unique element in the kernel gives injection:










h
(

g

1


)



=
h
(

g

2


)




⇔



h
(

g

1


)
⋅
h
(

g

2



)

−
1





=

e

H






⇔



h

(


g

1


∘

g

2


−
1



)




=

e

H


,

ker
⁡
(
h
)
=
{

e

G


}




⇒




g

1


∘

g

2


−
1





=

e

G






⇔




g

1





=

g

2








{\displaystyle {\begin{aligned}&&h(g_{1})&=h(g_{2})\\\Leftrightarrow &&h(g_{1})\cdot h(g_{2})^{-1}&=e_{H}\\\Leftrightarrow &&h\left(g_{1}\circ g_{2}^{-1}\right)&=e_{H},\ \operatorname {ker} (h)=\{e_{G}\}\\\Rightarrow &&g_{1}\circ g_{2}^{-1}&=e_{G}\\\Leftrightarrow &&g_{1}&=g_{2}\end{aligned}}}





Examples


Consider the cyclic group Z3 = (Z/3Z, +) = ({0, 1, 2}, +) and the group of integers (Z, +). The map h : Z → Z/3Z with h(u) = u mod 3 is a group homomorphism. It is surjective and its kernel consists of all integers which are divisible by 3.

The exponential map yields a group homomorphism from the group of real numbers R with addition to the group of non-zero real numbers R* with multiplication. The kernel is {0} and the image consists of the positive real numbers.
The exponential map also yields a group homomorphism from the group of complex numbers C with addition to the group of non-zero complex numbers C* with multiplication. This map is surjective and has the kernel {2πki : k ∈ Z}, as can be seen from Euler's formula. Fields like R and C that have homomorphisms from their additive group to their multiplicative group are thus called exponential fields.
The function



Φ
:
(

N

,
+
)
→
(

R

,
+
)


{\displaystyle \Phi :(\mathbb {N} ,+)\rightarrow (\mathbb {R} ,+)}

, defined by



Φ
(
x
)
=


2





x


{\displaystyle \Phi (x)={\sqrt[{}]{2}}x}

is a homomorphism.
Consider the two groups



(


R


+


,
∗
)


{\displaystyle (\mathbb {R} ^{+},*)}

and



(

R

,
+
)


{\displaystyle (\mathbb {R} ,+)}

, represented respectively by



G


{\displaystyle G}

and



H


{\displaystyle H}

, where





R


+




{\displaystyle \mathbb {R} ^{+}}

is the positive real numbers. Then, the function



f
:
G
→
H


{\displaystyle f:G\rightarrow H}

defined by the logarithm function is a homomorphism.




Category of groups


If h : G → H and k : H → K are group homomorphisms, then so is k ∘ h : G → K. This shows that the class of all groups, together with group homomorphisms as morphisms, forms a category (specifically the category of groups).




Homomorphisms of abelian groups


If G and H are abelian (i.e., commutative) groups, then the set Hom(G, H) of all group homomorphisms from G to H is itself an abelian group: the sum h + k of two homomorphisms is defined by

(h + k)(u) = h(u) + k(u) for all u in G.
The commutativity of H is needed to prove that h + k is again a group homomorphism.
The addition of homomorphisms is compatible with the composition of homomorphisms in the following sense: if f is in Hom(K, G), h, k are elements of Hom(G, H), and g is in Hom(H, L), then

(h + k) ∘ f = (h ∘ f) + (k ∘ f) and g ∘ (h + k) = (g ∘ h) + (g ∘ k).
Since the composition is associative, this shows that the set End(G) of all endomorphisms of an abelian group forms a ring, the endomorphism ring of G. For example, the endomorphism ring of the abelian group consisting of the direct sum of m copies of Z/nZ is isomorphic to the ring of m-by-m matrices with entries in Z/nZ. The above compatibility also shows that the category of all abelian groups with group homomorphisms forms a preadditive category; the existence of direct sums and well-behaved kernels makes this category the prototypical example of an abelian category.




See also


Homomorphism
Fundamental theorem on homomorphisms
Quasimorphism
Ring homomorphism




References


Dummit, D. S.; Foote, R. (2004). Abstract Algebra (3rd ed.). Wiley. pp. 71–72. ISBN 978-0-471-43334-7.
Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001




External links


Rowland, Todd & Weisstein, Eric W. "Group Homomorphism". MathWorld.

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