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

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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 C G . Show that if G/N is abelian for a normal subgroup N in G then N contains C G . (b) [15pts] Show that C G is normal in G . (c) [10pts] Find the commutator subgroup of Z n and A 5 (Hint: Part (b) and midterm 1, question 3).
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2. (a) [10pts] Center Z G of a group G consists of the elements of G that commute with every element of G . Find the center of D 2 n , the dihedral group with 2 n elements (or the symmetry group of a regular
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