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Handbook of mathematics for engineers and scienteists part 38

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Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 38', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 5.8. Some Facts from Group Theory 227 Theorem. The homomorphic image f G is a group. The image f e of the identity element e g G is the identity element of the group f G . Mutually inverse elements of G correspond to mutually inverse images in f G . Two groups G1 and G2 are said to be isomorphic if there exists a one-to-one mapping f of G1 onto G2 such that f ab f a f b for all a be. G1. Such a mapping is called an isomorphism or isomorphic mapping of the group G1 onto the group G2. Theorem. Any isomorphism of groups is invertible and the inverse mapping is also an isomorphism. An isomorphic mapping of a group G onto itself is called an automorphism of G. If f 1 G G and f2 G G are two automorphisms of a group G one can define another automorphism fi o f 2 G G by letting fi o f2 g fi f2 g for all g g G. This automorphism is called the composition of f1 and f2 and with this composition law the set of all automorphisms of G becomes a group called the automorphism group of G. 5.8.1-4. Subgroups. Cosets. Normal subgroups. Let G be a group. A subset G1 of the group G is called a subgroup if the following conditions hold 1. For any a and b belonging to G1 the product ab belongs to G1. 2. For any a belonging to G1 its inverse a 1 belongs to G1. These conditions ensure that any subgroup of a group is itself a group. Example 5. The identity element of a group is a subgroup. The subset of all even numbers is a subgroup of the additive group of all integers. The product of two subsets H1 and H2 of a group G is a set H3 that consists of all elements of the form h h2 where h1 H1 h2 e H2. In this case one writes H3 H H2. Let H be a subgroup of a group G and a some fixed element of G. The set aH is called a left coset and the set Ha is called a right coset of the subgroup H in G. Properties of left cosets right cosets have similar properties 1. If aeH then aH H. 2. Cosets aH and bH coincide if a 1 be H. 3. Two cosets of the same subgroup H either coincide or have no common .