92.421/521 Abstract Algebra

Fall 2011

Problem Set #3

Due October 31

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.    List the left and right cosets of the subgroups in each of the following.

(a)  (Subscript[A, n]) in (Subscript[S, n])	     (b)  ⟨5⟩ in (Subscript[ℤ, 20])

2.    Show that the additive group of real numbers has infinite index in the additive group of the complex numbers.

3.  Suppose that [G:H]=2.  If a and b are not in H, show that a b∈H.

4.  Prove that ℚ is not isomorphic to ℤ.

5.  Prove that G × H is isomorphic to H × G.

6.  Let (Subscript[H, 1]) and (Subscript[H, 2]) be subgroups of (Subscript[G, 1]) and (Subscript[G, 2]), respectively. Prove that (Subscript[H, 1])×(Subscript[H, 2]) 
is a subgroup of (Subscript[G, 1]) × (Subscript[G, 2]).