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 ![](Image463.gif)
- Show that the quotient group
is abelian.
- (G)
Show that
is minimal with respect to this property. That is, if
is abelian, then
(Hence ![](Image468.gif)
).
- Let
be a homomorphism of groups. Show that ker(
) is a normal subgroup of ![](Image471.gif)
- Let
be a group and
a subgroup of ![](Image428.gif)
.
- Show that if
, then
is normal in ![](Image476.gif)
- Given an example of groups
with
with
not normal in ![](Image480.gif)
- Let
be a group with center ![](Image482.gif)
.
- Show that
is normal in
.
- (G)
Show that
is cyclic ![](Image385.gif)
![](Image413.gif)
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 ![](Image492.gif)
.