Let G 0 -{(81%) = : a, b ER, a 0 (a) Show that G is a subgroup of SL(2). (b) For each of the following functions, determine whether it is a homomorphism, justifying your answer. (i) 1: (G, x) →→→ (G, X) a +0}. a 0 1/a (81%) → (¹0) b a (ii) 2: (G, x) →→→ (G, X) a 0 b 1/a a (1/a) → (6% 1/a) b b/a (iii) 03: (G, ×) → (R, +) → a-b (c) For any mapping o in part (b) that is a homomorphism, find its image and kernel.
Let G 0 -{(81%) = : a, b ER, a 0 (a) Show that G is a subgroup of SL(2). (b) For each of the following functions, determine whether it is a homomorphism, justifying your answer. (i) 1: (G, x) →→→ (G, X) a +0}. a 0 1/a (81%) → (¹0) b a (ii) 2: (G, x) →→→ (G, X) a 0 b 1/a a (1/a) → (6% 1/a) b b/a (iii) 03: (G, ×) → (R, +) → a-b (c) For any mapping o in part (b) that is a homomorphism, find its image and kernel.
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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Transcribed Image Text:Let
G
-{(81%)
1/a):
b
: a, b ER, a
(a) Show that G is a subgroup of SL(2).
(b) For each of the following functions, determine whether it is a
homomorphism, justifying your answer.
(i) 1 (G, x) → (G, X)
b
(ii) 2 (G, X) → (G, X)
1/a) → (¹1/⁰9)
0
a
a
b
+0}.
a
(8 1/a) → (b%a 1/a)
b
(iii) 03 (G, X)→ (R,+)
0
1/a) → a-
(c) For any mapping o in part (b) that is a homomorphism, find its image
and kernel.
(d) For any mapping o in part (b) that is a homomorphism, determine a
standard group that is isomorphic to the quotient group G/Ker o.
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