3330_Notes_3o6_fill

# 3330_Notes_3o6_fill - Math 3330 Section 3.6 Homomorphisms...

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Math 3330 Section 3.6 Homomorphisms Let G and G* be groups A map f is said to preserve the group operation if Isomorphisms preserve the group operation, and further indicate the algebraic structure of the two systems are equivalent. BUT, a map can preserve the group operations without being bijective. Definition: Homomorphism Let (, and be groups. ) G * G : ) ) A map is a homomorphism if for all x and y in G, * : fG G () ( ) ( fx y fx fy ⊗= : . Example: defined by : n fZ Z () [] fz z = for all integers z. Is f onto? Is f one-to-one?

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Example: Let be defined by : g ]] ( ) 6 gn n = . Is g a homomorphism? Is it onto? Is it one-to-one? Example: Define by : ( , ) ( , ) h + ×→ + _\ ( ) ln( ) hx x =
Images of Identities and Inverses

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3330_Notes_3o6_fill - Math 3330 Section 3.6 Homomorphisms...

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