Groups and Group Actions: Lecture 10

In which we think some more about homomorphisms, and meet normal subgroups.

• Proposition 45: Let Image may be NSFW.
  Clik here to view. be a homomorphism.  Then there is Image may be NSFW.
  Clik here to view. with Image may be NSFW.
  Clik here to view. for all Image may be NSFW.
  Clik here to view..  We proved this by defining Image may be NSFW.
  Clik here to view. and then using the fact that Image may be NSFW.
  Clik here to view. generates Image may be NSFW.
  Clik here to view..
• Proposition 46: Let Image may be NSFW.
  Clik here to view., Image may be NSFW.
  Clik here to view. be groups, let Image may be NSFW.
  Clik here to view. be a homomorphism.  Then
  1. Image may be NSFW.
     Clik here to view. is a subgroup of Image may be NSFW.
     Clik here to view.; and
  2. Image may be NSFW.
     Clik here to view. is a subgroup of Image may be NSFW.
     Clik here to view..
• We proved this using the subgroup test.
• Proposition 47: Let Image may be NSFW.
  Clik here to view., Image may be NSFW.
  Clik here to view.  be groups, let Image may be NSFW.
  Clik here to view. be a homomorphism.  Then Image may be NSFW.
  Clik here to view. is constant on each coset of Image may be NSFW.
  Clik here to view., and takes different values on different cosets.  We saw that Image may be NSFW.
  Clik here to view. if and only if Image may be NSFW.
  Clik here to view., and then used the coset equality test to see that this is equivalent to Image may be NSFW.
  Clik here to view..
• Corollary 48: Let Image may be NSFW.
  Clik here to view., Image may be NSFW.
  Clik here to view. be groups, let Image may be NSFW.
  Clik here to view. be a homomorphism.  Then Image may be NSFW.
  Clik here to view. is injective if and only if Image may be NSFW.
  Clik here to view..  This was immediate from Proposition 47, since Image may be NSFW.
  Clik here to view..
• Definition of a normal subgroup of a group.
• Definition of a simple group.
• Proposition 49: Let Image may be NSFW.
  Clik here to view., Image may be NSFW.
  Clik here to view. be groups, let Image may be NSFW.
  Clik here to view. be a homomorphism.  Then Image may be NSFW.
  Clik here to view. is a normal subgroup of Image may be NSFW.
  Clik here to view..  We already know from Proposition 46 that Image may be NSFW.
  Clik here to view. is a subgroup of Image may be NSFW.
  Clik here to view., so we just checked that if Image may be NSFW.
  Clik here to view. and Image may be NSFW.
  Clik here to view. then Image may be NSFW.
  Clik here to view..
• Proposition 50: Let Image may be NSFW.
  Clik here to view. be a subgroup of a group Image may be NSFW.
  Clik here to view. with index Image may be NSFW.
  Clik here to view..  Then Image may be NSFW.
  Clik here to view..  We argued that the only (left and right) cosets of Image may be NSFW.
  Clik here to view. in Image may be NSFW.
  Clik here to view. are Image may be NSFW.
  Clik here to view. and Image may be NSFW.
  Clik here to view., and used this to see that Image may be NSFW.
  Clik here to view. for all Image may be NSFW.
  Clik here to view..
• Definition of conjugacy classes.

Understanding today’s lecture

Pick some homomorphisms.  Can you identify their kernels and images?  Which homomorphisms are injective?  Which are surjective?

Pick a homomorphism Image may be NSFW.
Clik here to view. between two groups (pick explicit groups and an explicit homomorphism).  What are the cosets of Image may be NSFW.
Clik here to view.?  This might help you to get a feel for Proposition 47.

Can you prove Corollary 48 directly from the definitions, without using Proposition 47?

Pick some subgroups of groups.  Which are normal in their respective groups?

Further reading

Of course Wikipedia has a page about normal subgroups, and another about the notion of a simple group.  You will learn more about simple groups if you choose to study Group Theory next year.

MacTutor suggests that Galois was the first to recognise the significance of normal subgroups.

Preparation for Lecture 11

Can you show that if Image may be NSFW.
Clik here to view. is a normal subgroup of Image may be NSFW.
Clik here to view. then Image may be NSFW.
Clik here to view. (the set of left cosets of Image may be NSFW.
Clik here to view. in Image may be NSFW.
Clik here to view.) forms a group under the natural operation?