Math 420/620
Fall 1999
Problem Set 4
Problems denoted (G) are only required of the graduate students.
- 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
- Show that the quotient group is abelian.
- (G)
Show that is minimal with respect to this property. That is, if is abelian, then (Hence
).
- Let be a homomorphism of groups. Show that ker() is a normal subgroup of
- Let be a group and a subgroup of
.
- Show that if , then is normal in
- Given an example of groups with with not normal in
- Let be a group with center
.
- Show that is normal in .
- (G)
Show that is cyclic
is abelian.
- (G)
Show that every finitely generated subgroup of Q (the rational numbers) is cyclic, but that Q itself is not finitely generated.
- (G)
Let and be normal subgroups of with . Show that for all pairs and
.