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Let G be the set of all maps Tab: R -» R defined by Ta,b(x) = ax b, where a, b e R, a 0 (a) Show that Tab is a bijection (b) Show that T1,0 = Idr Find the explicit values of u, v in terms of a, b, c, d E R, a, c 0 (c) a,b -1 Find a formula for Tab (d) (e) (f) a,b Show that G is a group with the binary operation the composition of maps H={Ta,b» 'a,b, a 0 rational, b E R} (H, o) is also a group, nonabelian (g) К3{T,b:b € R} снCG (K, o) is an abelian group Notation: G-Aff(R)=R x R*: semiproduct For TabE G and v E H, Ta,b -1 (h) v Tа,ь Е Н a,b

Let G be the set of all maps Tab: R -» R defined by Ta,b(x) = ax b, where a, b e R, a 0 (a) Show that Tab is a bijection (b) Show that T1,0 = Idr Find the explicit values of u, v in terms of a, b, c, d E R, a, c 0 (c) a,b -1 Find a formula for Tab (d) (e) (f) a,b Show that G is a group with the binary operation the composition of maps H={Ta,b» 'a,b, a 0 rational, b E R} (H, o) is also a group, nonabelian (g) К3{T,b:b € R} снCG (K, o) is an abelian group Notation: G-Aff(R)=R x R*: semiproduct For TabE G and v E H, Ta,b -1 (h) v Tа,ь Е Н a,b

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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Let G be the set of all maps Tab: R -» R defined by Ta,b(x) = ax b, where a, b e R, a 0 (a) Show that Tab is a bijection (b) Show that T1,0 = Idr Find the explicit values of u, v in terms of a, b, c, d E R, a, c 0 (c) a,b -1 Find a formula for Tab (d) (e) (f) a,b Show that G is a group with the binary operation the composition of maps H={Ta,b» 'a,b, a 0 rational, b E R} (H, o) is also a group, nonabelian (g) К3{T,b:b € R} снCG (K, o) is an abelian group Notation: G-Aff(R)=R x R*: semiproduct For TabE G and v E H, Ta,b -1 (h) v Tа,ь Е Н a,b
Transcribed Image Text:Let G be the set of all maps Tab: R -» R defined by Ta,b(x) = ax b, where a, b e R, a 0 (a) Show that Tab is a bijection (b) Show that T1,0 = Idr Find the explicit values of u, v in terms of a, b, c, d E R, a, c 0 (c) a,b -1 Find a formula for Tab (d) (e) (f) a,b Show that G is a group with the binary operation the composition of maps H={Ta,b» 'a,b, a 0 rational, b E R} (H, o) is also a group, nonabelian (g) К3{T,b:b € R} снCG (K, o) is an abelian group Notation: G-Aff(R)=R x R*: semiproduct For TabE G and v E H, Ta,b -1 (h) v Tа,ь Е Н a,b
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