S. D. V. Stephens

notes / Studies in Algebra and Group Theory / chapter 1: Sets and Groups / From my notebooks

N6 Cosets, Lagrange’s Theorem and Normal Subgroups

Cosets

Definition N6.1 (Left coset). Given a subgroup of a group and , the set is the left coset of containing .

  • The relation is an equivalence relation on , whose equivalence classes are the left cosets.
  • The quotient set (the set of left cosets) is denoted . We have

Proof. That is an equivalence relation:

  • , therefore .
  • If , then , so , therefore .
  • If and , then , so , therefore .

That the classes are the left cosets: observe hence meow. ∎

Example N6.2. The cosets of are = the congruence classes mod (). So

Definition N6.3 (Index). The index of the subgroup is the number of cosets, .

Lagrange’s theorem

Theorem N6.4. When is a finite group, each coset has . Thus implies

Corollary N6.5 (Lagrange’s Theorem). If is a subgroup of a finite group , then divides .

Corollary N6.6. For all , the order of (the minimal with ) equals , which divides .

Corollary N6.7. If , where is prime, then .

Proof. Let with . Then has order , therefore generates all of : , and via . ∎

Right cosets vs. left cosets

Also let : this is a subgroup of (conjugation by ).

(“Awooga”: Auroux when he sees .)

Definition N6.8 (Normal subgroup). is a normal subgroup if it is a subgroup and for all ( and are equal, i.e. ).

Equivalently: for all (from , right-multiply by ), i.e. is closed under conjugation by all elements of .

Group structure on

  • For , the set of cosets inherits a group structure because is compatible with : , so .
  • In general, trying to define a group structure on by does NOT work. The issue: assume (), and the same with . Does it then follow that , i.e. ? Girl, of course not (proof by “prof said”).

Example N6.9. Let and consider the subgroup . Notice (i.e. ) (????). Then , but does it follow that ? Nope! We have , while (the notes check this by drawing where each sends a labelled square), so .

However! The property holds when is the kernel of some homomorphism , : Being in the same coset = having the same : .

  • If and , then and , so is , which is ; hence . So we can define a group operation on by . In fact we have a bijection This is a group isomorphism! .

Examples of normal subgroups

  • Any subgroup of an abelian group is normal.
  • In , is NOT normal ().

Theorem N6.10. Given a group and a subgroup , the following are equivalent:

  1. There exists a group homomorphism with .
  2. is a normal subgroup.
  3. has a group structure given by .

Proof. (i) (ii) (we have already seen (i) (iii)). Suppose . We want to compare right and left cosets (). For all , so , and is normal.

(ii) (iii). Assume is normal. Then and : trivial. ∎