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. Suppose that G and H are groups, and that φ : G → H is a homomorphism. Prove that G/Ker(φ)-φ(G). (The notation įs shorthand for "is isomorphic to.) Definition: Suppose G is a group, and H a normal subgroup of G. The group consisting of the set G/H with operation defined by (aH (bH) (ab) is called the quotient group of G by H. (Sometime the ter factor group" is used in place of "quotient group".) Definitions: . Suppose that (Ga) and (Ho) are groups. We say that a function φ : G → H is a homomorphism if for all a, be G, ф(a b)-ф(a)od(b). Definition: We say that groups are isomorphic if there exists an isomorphism between them. If G and H are isomorphic groups, we write in symbols GH exists an isomorpiism

. Suppose that G and H are groups, and that φ : G → H is a homomorphism. Prove that G/Ker(φ)-φ(G). (The notation įs shorthand for "is isomorphic to.) Definition: Suppose G is a group, and H a normal subgroup of G. The group consisting of the set G/H with operation defined by (aH (bH) (ab) is called the quotient group of G by H. (Sometime the ter factor group" is used in place of "quotient group".) Definitions: . Suppose that (Ga) and (Ho) are groups. We say that a function φ : G → H is a homomorphism if for all a, be G, ф(a b)-ф(a)od(b). Definition: We say that groups are isomorphic if there exists an isomorphism between them. If G and H are isomorphic groups, we write in symbols GH exists an isomorpiism

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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. Suppose that G and H are groups, and that φ : G → H is a homomorphism. Prove that G/Ker(φ)-φ(G). (The notation įs shorthand for "is isomorphic to.) Definition: Suppose G is a group, and H a normal subgroup of G. The group consisting of the set G/H with operation defined by (aH (bH) (ab) is called the quotient group of G by H. (Sometime the ter factor group" is used in place of "quotient group".) Definitions: . Suppose that (Ga) and (Ho) are groups. We say that a function φ : G → H is a homomorphism if for all a, be G, ф(a b)-ф(a)od(b). Definition: We say that groups are isomorphic if there exists an isomorphism between them. If G and H are isomorphic groups, we write in symbols GH exists an isomorpiism
Transcribed Image Text:. Suppose that G and H are groups, and that φ : G → H is a homomorphism. Prove that G/Ker(φ)-φ(G). (The notation įs shorthand for "is isomorphic to.) Definition: Suppose G is a group, and H a normal subgroup of G. The group consisting of the set G/H with operation defined by (aH (bH) (ab) is called the quotient group of G by H. (Sometime the ter factor group" is used in place of "quotient group".) Definitions: . Suppose that (Ga) and (Ho) are groups. We say that a function φ : G → H is a homomorphism if for all a, be G, ф(a b)-ф(a)od(b). Definition: We say that groups are isomorphic if there exists an isomorphism between them. If G and H are isomorphic groups, we write in symbols GH exists an isomorpiism
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