7. Let (G, *) and (K, o) be groups. Let ø: G → K be a group homomorphism (not neces- sarily an isomorphism). Prove that (a) $(eg) = eK, so O maps the identity of G into the identity of K. (b) Show that for all g E G, ø(g¬1) = [ø(g)]¬1. That is, o maps the inverse of g into the inverse of ø(g).
7. Let (G, *) and (K, o) be groups. Let ø: G → K be a group homomorphism (not neces- sarily an isomorphism). Prove that (a) $(eg) = eK, so O maps the identity of G into the identity of K. (b) Show that for all g E G, ø(g¬1) = [ø(g)]¬1. That is, o maps the inverse of g into the inverse of ø(g).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![7. Let (G, *) and (K, o) be groups. Let ø: G → K be a group homomorphism (not neces-
sarily an isomorphism). Prove that
(a) $(eg) = eK, so o maps the identity of G into the identity of K.
(b) Show that for all g e G, $(g-1) = [#(g)]¬1.
That is, ø maps the inverse of g into the inverse of ø(g).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7abbf2d5-6ef3-489c-b2da-4d9327a88b2a%2Fc747ca72-9d56-4235-834b-1beea7de97a8%2F0qwyvaq_processed.png&w=3840&q=75)
Transcribed Image Text:7. Let (G, *) and (K, o) be groups. Let ø: G → K be a group homomorphism (not neces-
sarily an isomorphism). Prove that
(a) $(eg) = eK, so o maps the identity of G into the identity of K.
(b) Show that for all g e G, $(g-1) = [#(g)]¬1.
That is, ø maps the inverse of g into the inverse of ø(g).
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