词条 | Group homomorphism |
释义 |
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 where the group operation on the left hand side of the equation is that of G and on the right hand 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, and it also maps inverses to inverses in the sense that Hence one can say that h "is compatible with the group structure". Older notations for the homomorphism h(x) may be xh or xh, though this may be confused as an index or a general subscript. A more recent trend is to write group homomorphisms on the right of their arguments, omitting brackets, so that h(x) becomes simply x h. This approach is especially prevalent in areas of group theory where automata play a role, since it accords better with the convention that automata read words from left to right. 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. IntuitionThe purpose of defining a group homomorphism is to create functions that preserve the algebraic structure. An equivalent definition of group homomorphism is: The function h : G → H is a group homomorphism if whenever a ∗ b = c we have h(a) ⋅ h(b) = h(c). In other words, the group H in some sense has a similar algebraic structure as G and the homomorphism h preserves that. Types of group homomorphism
Image and kernel{{main article|Image (mathematics)|kernel (algebra)}}We define the kernel of h to be the set of elements in G which are mapped to the identity in H and the image of h to be 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 and the image of h is a subgroup of H: If and only if {{nowrap|ker(h) {{=}} {eG}}}, the homomorphism, h, is a group monomorphism; i.e., h is injective (one-to-one). Injection directly gives that there is a unique element in the kernel, and a unique element in the kernel gives injection: Examples
Consider the group For any complex number u the function fu : G → C defined by: is a group homomorphism. | Consider multiplicative group of positive real numbers (R+, ⋅) for any complex number u the function fu : R+ → C defined by: is a group homomorphism. }}
The category of groupsIf {{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. Homomorphisms of abelian groupsIf G and H are abelian (i.e., commutative) groups, then the set {{nowrap|Hom(G, H)}} of all group homomorphisms from G to H is itself an abelian group: the sum {{nowrap|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 {{nowrap|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 {{nowrap|Hom(K, G)}}, h, k are elements of {{nowrap|Hom(G, H)}}, and g is in {{nowrap|Hom(H, L)}}, then {{nowrap|1=(h + k) ∘ f = (h ∘ f) + (k ∘ f)}} and {{nowrap|1=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{{Div col}}
References
| last1 = Dummit | first1 = D. S. | last2 = Foote | first2 = R. | title = Abstract Algebra | publisher = Wiley | pages = 71–72 | year = 2004 | edition = 3rd | isbn = 978-0-471-43334-7 }}
External links
2 : Group theory|Morphisms |
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