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group homomorphism preserves identity i.e. for any homomorphism, f :G → G' , if e is the identity element of G then f(e) is the identity element of G'.

group homomorphism preserves identity i.e. for any homomorphism, f :G → G' , if e is the identity element of G then f(e) is the identity element of G'.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Please proof

My textbook (Herstein) proves that a group homomorphism preserves identity i.e. for any homomorphism, f :G → G' , if e is the identity element of G then f(e) is the identity element of G! Here is the proof: x = xe → f(xe) = f(x)f(e) = f(x) Vx E G I understand the proof, but it seems as though it doesn't completely prove the statement. For example, if f is not onto, then there exists an element y in G' such that f (x) # y for all x in G. The proof above doesnt show that ye' = yf(e) = y which would need to still be the case in order for f(e) to be the identity element of G'.
Transcribed Image Text:My textbook (Herstein) proves that a group homomorphism preserves identity i.e. for any homomorphism, f :G → G' , if e is the identity element of G then f(e) is the identity element of G! Here is the proof: x = xe → f(xe) = f(x)f(e) = f(x) Vx E G I understand the proof, but it seems as though it doesn't completely prove the statement. For example, if f is not onto, then there exists an element y in G' such that f (x) # y for all x in G. The proof above doesnt show that ye' = yf(e) = y which would need to still be the case in order for f(e) to be the identity element of G'.
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