S. D. V. Stephens

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

N7 Exact Sequences, the Symmetric Group and Commutators

Review

  • Cosets of : .
  • Normal subgroup: .
  • (a quotient group iff is normal).

Definition N7.1 (Simple group). A group is a simple group if its normal subgroups are and .

Short exact sequences

Definition N7.2 (Short exact sequence). A short exact sequence of groups is a diagram of groups mapping to each other by homomorphisms, such that the image of each homomorphism equals the kernel of the next one. (So is trivial, .)

Unpacking this:

  • , so is injective.
  • : for all , for some .
  • , so is surjective.

Consequently:

  1. is isomorphic via to a subgroup , a normal subgroup of .
  2. , , is a group isomorphism.

So the sequence looks like (girl what?)

Example N7.3.

  1. , with Here (a normal subgroup) is isomorphic to , and the quotient is therefore (how??).
  2. , with (the -cycles), and the second map “the sign homomorphism”. So also contains a normal subgroup with quotient . Am lost :(

More about

Definition N7.4 (Cycle). A cycle (-cycle) is the permutation that maps and all others to themselves. A -cycle is a transposition.

Proposition N7.5. Every permutation can be expressed as a product of disjoint cycles (cycling disjoint subsets of ), uniquely up to reordering the factors. Disjoint cycles commute.

Example N7.6.

  • A -cycle can be written as a product of transpositions: So is generated by the transpositions , .
  • Also: for example . Girl what? Girl yassss! Yassen queenen (little German boy). (Margin: parity: or .) [Bubble sort algorithm.]

Permutations are odd or even depending on the length of an expression of as a product of transpositions the parity of

Proposition N7.7 (Claim). Even permutations form a normal subgroup , called the alternating subgroup of :

. has a normal subgroup (?), but for , is simple.

There is no formula for the roots of polynomials of degree : Galois theory (Math 123 and 129).

Noncommutativity of a group

Definition N7.8 (Center). The center is the set of all elements that commute with everyone (the “central elements”):

If , then for all , so . This is a normal abelian subgroup, and is abelian.

Definition N7.9 (Commutator subgroup). The commutator subgroup is It is always a normal subgroup.

Example N7.10. , and iff is abelian.

The quotient is called the abelianization of . Writing : since .