2. Let ø : G → G' be a homomorphism of groups, and let H be a subgroup of G.(a) Let a E G. Prove that$(a") = 4(a)" for all n e Z.(b) Prove that if H is cyclic, then ø[H] is also cyclic.(c) Prove that if H is normal in G and ø is onto, then ø[H]is normal in G'.

Given ϕ:G -> G' is a homomorphism. H is a subgroup of G. The solution to the subparts is given…

2. Let o : G → G' be a homomorphism of groups, and let H be a subgroup of G. (a) Let a E G. Prove that $(a") = ¢(a)" for all n e Z. (b) Prove that if H is cyclic, then o[H] is also cyclic. (c) Prove that if H is normal in G and o is onto, then [H] is normal in G'.

2. Let o : G → G' be a homomorphism of groups, and let H be a subgroup of G. (a) Let a E G. Prove that $(a") = ¢(a)" for all n e Z. (b) Prove that if H is cyclic, then o[H] is also cyclic. (c) Prove that if H is normal in G and o is onto, then [H] is normal in 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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Question

2. Let ø : G → G' be a homomorphism of groups, and let H be a subgroup of G.
(a) Let a E G. Prove that
$(a") = 4(a)" for all n e Z.
(b) Prove that if H is cyclic, then ø[H] is also cyclic.
(c) Prove that if H is normal in G and ø is onto, then ø[H]is normal in G'.

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