m321s2

m321s2 - B U Department of Mathematics Fall 2008 Math 321 -...

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Unformatted text preview: B U Department of Mathematics Fall 2008 Math 321 - Algebra, Second Midterm Exam, 16/11/2008, 13:00-15:00 Full Name Student ID : : over 100 1. (a) [10pts] The set of all elements aba−1 b−1 for a, b ∈ G generates a subgroup of G. This subgroup is called the commutator subgroup; denote it by CG . Show that if G/N is abelian for a normal subgroup N in G then N contains CG . (b) [15pts] Show that CG is normal in G. (c) [10pts] Find the commutator subgroup of Zn and A5 (Hint: Part (b) and midterm 1, question 3). 2. (a) [10pts] Center ZG of a group G consists of the elements of G that commute with every element of G. Find the center of D2n , the dihedral group with 2n elements (or the symmetry group of a regular 2n-gon). (b) [15pts] An inner automorphism of a group G is an automorphism that sends each element to its conjugate under a fixed element of G. Without looking at inner automorphisms of D2n explicitly, determine the number of inner automorphisms of D2n . 3. (a) [0pts] Write down a homomorphism from Z10 onto Z2 . (b) [20pts] Suppose there is a homomorphism from a group G onto Z10 . Prove that G has normal subgroups of index 2 and 5. (Hint: Use a fundamental theorem about homomorphisms.) 4. [20pts] In a group of order 33, how many elements are of order 11? ...
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This note was uploaded on 05/04/2011 for the course MATH 321 taught by Professor Talinbudak during the Spring '11 term at BU.

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m321s2 - B U Department of Mathematics Fall 2008 Math 321 -...

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