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(7) Let (x, y) € R × R. Consider the following functions from R² to R² a((x, y)) = (-x, y) and b(x,y)) = (x, -y). (a) What is the order of a? of b? Note: a" = a o a o ao... a. n times (b) Let f be of the form a'bi where 0

(7) Let (x, y) € R × R. Consider the following functions from R² to R² a((x, y)) = (-x, y) and b(x,y)) = (x, -y). (a) What is the order of a? of b? Note: a" = a o a o ao... a. n times (b) Let f be of the form a'bi where 0

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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plz solve the question 20-30 min max urgently with explanations and i will give you multiple upvote.
(7) Let (x, y) E R x R. Consider the following functions from R? to R? a((x, y)) = (-x, y) and b(x, y)) = (x, -y). (a) What is the order of a? of b? Note: a" = a o a o ao... o a. n times (b) Let f be of the form a'bi where 0 <is la| and 0 s js [b|. Fill in the table with these distinct functions and their images. image of (x, y) f (x, y) a It may be nice to enumerate the elements alphabetically, e.g. list ab ahead of a b. (c) The set K = { a'bi | 0<is la| and 0 sjs lb| } is a group. Construct the complete group table for K. Bonus: It is optional to answer this or not. (d) Let G = {1,2, 3, 4, 5, 6, 7, 8 } be a group whose proper subgroups are A = {1}, B = {1,3}, C = {1,5}, D = {1,6}, E = {1,7}, F = {1,8}, H = {1,2, 3, 4}, I = {1,3,5,7}, and J = {1,3,6, 8}. Determine a correspondence x+ y between the elements x of K and y of G such that K = G. %3D (e) Draw the lattice of subgroups diagram of G and K to reveal the isomorphic structure of the two groups.
Transcribed Image Text:(7) Let (x, y) E R x R. Consider the following functions from R? to R? a((x, y)) = (-x, y) and b(x, y)) = (x, -y). (a) What is the order of a? of b? Note: a" = a o a o ao... o a. n times (b) Let f be of the form a'bi where 0 <is la| and 0 s js [b|. Fill in the table with these distinct functions and their images. image of (x, y) f (x, y) a It may be nice to enumerate the elements alphabetically, e.g. list ab ahead of a b. (c) The set K = { a'bi | 0<is la| and 0 sjs lb| } is a group. Construct the complete group table for K. Bonus: It is optional to answer this or not. (d) Let G = {1,2, 3, 4, 5, 6, 7, 8 } be a group whose proper subgroups are A = {1}, B = {1,3}, C = {1,5}, D = {1,6}, E = {1,7}, F = {1,8}, H = {1,2, 3, 4}, I = {1,3,5,7}, and J = {1,3,6, 8}. Determine a correspondence x+ y between the elements x of K and y of G such that K = G. %3D (e) Draw the lattice of subgroups diagram of G and K to reveal the isomorphic structure of the two groups.
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