Group homomorphism: Difference between revisions

[[Image:Group homomorphism ver.2.svg|right|thumb|250px|Depiction of a group homomorphism (''h'') from ''G'' (left) to ''H'' (right). The oval inside ''H'' is the [[Image_(mathematics)|image]] of ''h''. ''N'' is the [[Kernel_(algebra)#Group_homomorphisms|kernel]] of ''h'' and ''aN'' is a [[coset]] of ''N''.]]

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 group]]s is often required to be continuous.

Let <math>e_{H}</math> be the identity element of the (''H'', ·) group and <math>u \in G</math>, then  

Now by multiplying for the inverse of <math>h(u)</math> (or applying the cancellation rule) we obtain  

Therefore for the uniqueness of the inverse: <math>h(u^{-1}) =  h(u)^{-1}</math>.

{{main article|Image (mathematics)|kernel (algebra)}}

We define the ''[[kernel (algebra)|kernel]] of h'' to be the set of elements in ''G'' which are mapped to the identity in ''H''

The kernel of h is a [[normal subgroup]] of ''G''. Assume <math>u \in \operatorname{ker}(h)</math> and show <math>g^{-1} \circ u \circ g \in \operatorname{ker}(h)</math> for arbitrary <math>u, g</math>:

The image of h is a [[subgroup]] of ''H''.

The homomorphism, ''h'', is a [[#monomorphism|''group monomorphism'']]; i.e., ''h'' is injective (one-to-one) if and only if {{nowrap|ker(''h'') {{=}} {''e''_''G''}}}. Injection directly gives that there is a unique element in the kernel, and, conversely, a unique element in the kernel gives injection:

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

forms a group under matrix multiplication. For any complex number ''u'' the function ''f_u'' : ''G'' → '''C^*''' defined by

Consider a multiplicative group of [[positive real numbers]] ('''R'''⁺, ⋅) for any complex number ''u''. Then the function ''f_u'' : '''R'''⁺ → '''C''' defined by

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

*{{MathWorld|title=Group Homomorphism|urlname=GroupHomomorphism|author=Rowland, Todd|author2=Weisstein, Eric W.|name-list-style=amp}}