HW3.F09 150A hass - Math 150A Homework 3 Selected...

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Unformatted text preview: Math 150A - Homework 3- Selected Solutions (Based on solutions prepared by Jeff Ferreira ) 2.3.2 Prove that the products ab and ba are conjugate elements in a group G . Proof. Wee need to show the existence of an element c ∈ G such that ab = c ( ba ) c- 1 . Taking c = a we see ab = a ( ba ) a- 1 . 2.3.6a Let φ : G → G be an isomorphism of groups, let x, y ∈ G , and let x = φ ( x ), y = φ ( y ). Prove that the orders of x and x are equal. Proof. Let n ≥ 1 be the order of x . This means precisely that x n = 1 and x p 6 = 1 for 0 < p < n . Now apply φ to these two conditions, and we see that x n = φ ( x ) n = φ ( x n ) = 1 and for 0 < p < n we must have x p = φ ( x p ) 6 = 1, since φ is one-to-one. So n is the order of x , too. 2.3.8 Prove that the matrices 1 2 and 1 3 2 are conjugate in GL 2 ( R ). Proof. Consider the matrix 1 3 1 . Then computation yields 1 3 1 1 2 1 3 1- 1 = 1 3 2 Note that 1 3 1- 1 = 1- 3 1 It’s also okay to conjugate the other matrix....
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HW3.F09 150A hass - Math 150A Homework 3 Selected...

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