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1- Let G = (U15, ×), and let N = <2>, the subspace generated by 2. List all the right cosets of N, and find the order of each one of them in G/N. %3D 2- If G is cyclic group, and N is any subgroup of G, prove that G/N is also cyclic. 3- Let G be a group and let p and q be two prime numbers with p q. Show that if H and K are two subgroups of G with |H| = p and |K| = q then HOK = {e}.

1- Let G = (U15, ×), and let N = <2>, the subspace generated by 2. List all the right cosets of N, and find the order of each one of them in G/N. %3D 2- If G is cyclic group, and N is any subgroup of G, prove that G/N is also cyclic. 3- Let G be a group and let p and q be two prime numbers with p q. Show that if H and K are two subgroups of G with |H| = p and |K| = q then HOK = {e}.

Advanced Engineering Mathematics
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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1- Let G = (U15, ×), and let N =<2>, the subspace generated by 2. List all the right cosets of N, and find the order of each one of them in G/N. 2- If G is cyclic group, and N is any subgroup of G, prove that G/N is also cyclic. 3- Let G be a group and let p and q be two prime numbers with p + q. Show that if H and K are two subgroups of G with |H|=p and |K| = q then HOK = {e}.
Transcribed Image Text:1- Let G = (U15, ×), and let N =<2>, the subspace generated by 2. List all the right cosets of N, and find the order of each one of them in G/N. 2- If G is cyclic group, and N is any subgroup of G, prove that G/N is also cyclic. 3- Let G be a group and let p and q be two prime numbers with p + q. Show that if H and K are two subgroups of G with |H|=p and |K| = q then HOK = {e}.
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