Math 420/620

Fall 1999

Problem Set 4

 

Problems denoted (G) are only required of the graduate students.

 

  1. Given a group we recall that a commutator is an element of the form with and that , the commutator subgroup, is a normal subgroup of
  1. Show that the quotient group is abelian.
  2. (G) Show that is minimal with respect to this property. That is, if is abelian, then (Hence
).

 

  1. Let be a homomorphism of groups. Show that ker() is a normal subgroup of

    2.  

  2. Let be a group and a subgroup of
.
  1. Show that if , then is normal in
  2. Given an example of groups with with not normal in

 

  1. Let be a group with center
.
  1. Show that is normal in .
  2. (G) Show that is cyclic
is abelian.

 

  1. (G) Show that every finitely generated subgroup of Q (the rational numbers) is cyclic, but that Q itself is not finitely generated.

    2.  

  2. (G) Let and be normal subgroups of with . Show that for all pairs and
.