This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: B U Department of Mathematics
Fall 2008 Math 321  Algebra, Second Midterm Exam, 16/11/2008, 13:0015:00 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 2ngon). (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? ...
View
Full
Document
This note was uploaded on 05/04/2011 for the course MATH 321 taught by Professor Talinbudak during the Spring '11 term at BU.
 Spring '11
 talinbudak
 Algebra

Click to edit the document details