midterm2_study

midterm2_study - Study sheet for Math 4107 midterm 2, Fall...

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Unformatted text preview: Study sheet for Math 4107 midterm 2, Fall 2009 November 10, 2009 Know all the material covered up to and including the first midterm. Know what an automorphism and inner-automorphism are, and know that they form a group under composition. Know that the group of inner-automorphisms on a group G is isomorphic to G/Z , where Z is the center. Know how to show that every finite group G of order greater than 2 (including now groups of even order) has a non-trivial automorphism. Know that Aut(( Z /n Z ) + ) is isomorphic to ( Z /n Z ) * . Know the definition of S n and A n (as I covered in class S n is the sym- metric group on { 1 , 2 ,...,n } , while A n is the corresponding alternating group). Know how to write every permutation as a product of disjoint cycles, and know how to use cycle notation. Know that every element of A n is a product of 3-cycles. Know the general form of the conju- gate of a permutation when written in disjoint cycle form it basically has the same cycle structure. Know how many equivalence classes ofhas the same cycle structure....
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midterm2_study - Study sheet for Math 4107 midterm 2, Fall...

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