Homework1Solutions

Homework1Solutions - Homework 1 Solutions Math 332, Spring...

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Math 332, Spring 2010 Problem 1. Let G be a group. Two elements g,h G are said to be conjugate if there exists an element a G for which h = a - 1 ga . (a) Proposition. Conjugacy is an equivalence relation on G . Proof. We must prove that conjugacy is reflexive, symmetric, and transitive. Reflexive If g G , then g = e - 1 ge , and therefore g is conjugate to g . Symmetric Let g,h G and suppose that g is conjugate to h . Then g = a - 1 ha for some element a G . Solving the above equation for h yields h = aga - 1 = b - 1 gb, where b = a - 1 . This proves that h is conjugate to g , and therefore conjugacy is symmetric. Transitive Let g,h,k G and suppose that g is conjugate to h and h is conjugate to k . Then g = a - 1 ha and h = b - 1 kb for some elements a,b G . Substituting the second equation into the first yields g = a - 1 b - 1 kba = c - 1 kc, where c = ba . This proves that g is conjugate to k , and therefore conjugacy is transitive. 1
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Homework1Solutions - Homework 1 Solutions Math 332, Spring...

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