Math 561 H Fall 2011 Homework 1 Drew Armstrong 1. Let G be a group. Prove that the identity element of G is unique (that is, there is only one element of G satisfying the deﬁning property of an identity element). This justiﬁes our use of the special symbol “ e ”. 2. Let G be a ﬁnite group. (a) Show that there are an odd number of x ∈ G such that x 3 = e . (b) Show that there are an even number of x ∈ G such that x 2 6 = e . (Hint: What is the inverse of x n ?) 3. Let G be a ﬁnite group. For all a,b ∈ G , show that ab and ba have the same order as elements of G . 4. Let G be a group and ﬁx an element g ∈ G . Deﬁne a function φ g : G → G by φ g ( h ) = ghg-1 for all h ∈ G . (a) Prove that φ g is a bijection (one-to-one and onto). (b) Prove that φ g ( ab ) = φ g ( a ) φ g ( b ) for all a,b ∈ G . These two properties mean that φ g is an automorphism (a “symmetry”) of G . 5.
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This note was uploaded on 01/08/2012 for the course MATH 561 taught by Professor Armstrong during the Spring '11 term at University of Miami.