Math 720
Fall 2000
Homework 3
- Show that any finite
group, G, can be embedded in Sn (that is, is a subgroup of Sn)
for some n.
- Compute:
a)
All
normal subgroups of Dn.
b)
A
subgroup of order 20 in S5.
c)
Is
there a subgroup of order 40 in S5?
- Show that Sn
is generated by two elements.
- Show that S3
is not the direct product of any family of its proper subgroups.
- Let G be abelian with
subgroups H and K. Show that
if and only if
there exist homomorphisms:
![](./M720.3.F2000_files/image004.gif)
Such that
·
![](./M720.3.F2000_files/image006.gif)
·
![](./M720.3.F2000_files/image008.gif)
·
![](./M720.3.F2000_files/image010.gif)
·
![](./M720.3.F2000_files/image012.gif)
- It is true that if H
and G are groups and
and
Then
Is it true that if H and G are groups with
and G isomorphic
to a subgroup of H is it true that ![](./M720.3.F2000_files/image022.gif)