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The kernel is a subgroup of G and the image is a subgroup of H. |
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The kernel is a normal subgroup of G (in fact, h(g-1 u g) = h(g)-1 eH h(g) = h(g)-1 h(g) = eH) and the image is a subgroup of H. |
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* Consider the cyclic group Z3 = {0, 1, 2} and the group of integers Z with addition. The map h: Z -> Z3 with h(u) = u modulo 3 is a group homorphism (see modular arithmetic). It is surjective and its kernel consists of all integers which are divisible by 3. |
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* Consider the cyclic group Z3 = {0, 1, 2} and the group of integers Z with addition. The map h : Z -> Z3 with h(u) = u modulo 3 is a group homorphism (see modular arithmetic). It is surjective and its kernel consists of all integers which are divisible by 3. |
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* Given any two groups G and H, the map h : G -> H which sends every element of G to the identity element of H is a homomorphism; its kernel is all of G. * Given any group G, the identity map id : G -> G with id(u) = u for all u in G is a group homorphism. |
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If h : G -> H and k : H -> K are group homomorphisms, then so is k o h : G -> K. This shows that the class of all groups, together with group homomorphisms as morphisms, form a category. |
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If h : G -> H and k : H -> K are group homomorphisms, then so is k o h : G -> K. This shows that the class of all groups, together with group homomorphisms as morphisms, forms a category. |
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If h: G -> G is a group homomorphism, we call it an endomorphism. If furthermore it is bijective and hence an isomorphism, it is called an automorphism. The set of all automorphisms of a group G, with functional composition as operation, forms itself 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 and multiplication with -1; it is isomorphic to Z2. |
If h: G -> G is a group homomorphism, we call it an endomorphism of G. If furthermore it is bijective and hence an isomorphism, it is called an automorphism. The set of all automorphisms of a group G, with functional composition as operation, forms itself 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 and multiplication with -1; it is isomorphic to Z2.Homorphisms of abelian groupsIf 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) o f = (h o f) + (k o f) and g o (h + k) = (g o h) + (g o k). 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 two copies of Z2 is isomorphic to the ring of 2-by-2 matrices with entries in Z2. The above compatibility also shows that category of all abelian groups with group homomorphisms forms an [additive category]?; the existence of kernels makes this category the prototypical example of an [abelian category]?. |
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