92.421/521 Abstract Algebra

Fall 2011

Problem Set #4

Due November 14

Instructions:   Do each problem on a separate sheet of paper.  On each sheet, write your name And problem statement (it can be abbreviated). Include all logical steps/observations.

1.  Which of the following subgroups of (Subscript[D, 6]) are normal?

(a)  ⟨s⟩     (b)    ⟨r⟩     (c)   ⟨(r^2)⟩

Use the representation of (Subscript[D, 6]) where  |s|=2, |r|=6, and  s r = (r^5)s.

2.  If a group G has exactly one subgroup H of order k, prove that H is normal in G.

3.  (a)  Show that a homomorphism defined on a cyclic group is completely determined by its action on the generator of the group.

(b)   Describe all homomorphisms from

(i)   (Subscript[ℤ, 20]) into (Subscript[ℤ, 18])

(i)   ℤ  into  (Subscript[ℤ, 12]).

We’ll do this in class tonight

Define the centralizer of an element g in a group G to be the set

C(g) ={x∈G | x g =g x}

Show that C(g) is a subgroup of G.  If g generates a normal subgroup of G, prove that C(g) is normal in G