S. D. V. Stephens

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

5 Exact Sequences, Permutations, Free Groups; Rings and Fields

· Denis Auroux

Original Professor Auroux’s handwritten notes (PDF), behind a password for now:

Last time: cosets ; normal subgroups (, i.e. ); the quotient group , defined if is normal. Today: miscellaneous things about groups, then on to rings and fields.

Simple groups and short exact sequences

Definition 5.1 (Simple group). Say a group is simple if it has no normal subgroups other than and .

We use normal subgroups to view as built from hopefully simpler groups and . Simple groups are then the basic building blocks.

Definition 5.2 (Short exact sequence). A short exact sequence of groups is a diagram

of groups mapped to each other by homomorphisms (trivial at the start and end) such that the image of each homomorphism equals the kernel of the next one. (One can also consider longer exact sequences.)

In a short exact sequence:

  • , i.e. is injective;
  • (so: , for some );
  • , i.e. is surjective.

In other terms:

  1. is isomorphic (via ) to , a normal subgroup of (kernels are normal), and
  2. (the cosets of ; is a group because is normal) is isomorphic to via

Example 5.3. Here is a normal subgroup , with quotient .

Example 5.4. There exists an exact sequence ( also contains a normal subgroup , namely , with quotient .)

So the first and third terms of a short exact sequence don’t determine the second.

More about

Definition 5.5 (Cycle). A cycle , where are distinct elements of , is the permutation mapping , , …, , and all other elements to themselves.

Proposition 5.6. Any permutation can be expressed as a product of disjoint cycles, uniquely up to reordering the factors. (Disjoint cycles commute, so the order doesn’t matter.)

Example 5.7. The cycle lists the successive images of under until it returns to ; then do the same for the other elements not in the previous cycles, which gives .

A -cycle can be written as a product of transpositions (= 2-cycles):

So is generated by the transpositions , . In fact it is generated by . (We can write in terms of those: , plus induction on .) [See also: the bubble sort algorithm.]

Definition 5.8 (Even and odd permutations; the alternating group). Permutations are odd or even depending on the length of an expression of as a product of transpositions ( the parity of ).

The even permutations form a normal subgroup , the alternating group :

Note. That parity is well defined is nontrivial! One needs to show that multiplication by a transposition always changes the parity.

Theorem 5.9 (Fact). , and has a normal subgroup , but for , is simple, i.e. its only normal subgroups are and itself.

Remark (Advertisement for Galois theory (Math 123)). This fact is used to prove that there is no general formula for solving polynomial equations of degree ! The quadratic formula has a , and the sign is there because over there’s not a consistent choice of of all complex numbers; the ambiguity is in permuting the two roots. The Cardano formula for cubics has in it. The and ambiguities in choosing these roots combine to an permuting the roots. Similarly, the formula for roots of a degree 5 equation should have a built-in symmetry, but any expression involving will have symmetry group built from cyclic ’s. This can’t be since is simple.

Two constructions that measure non-commutativity

Two constructions help understand the extent of non-commutativity in a group.

Definition 5.10 (Center). The center of is .

Since elements of the center commute with everyone, they commute with each other, so is abelian! Also, , so is a normal subgroup of . is abelian iff .

Definition 5.11 (Commutator subgroup, abelianization). The commutator subgroup is where (the “commutator” of and ; it equals iff ).

The quotient is called the abelianization of .

This is a normal subgroup because so for all .

contains all commutators , i.e. all commutators lie in the coset of the identity, hence all commutators in are equal to the identity: for all .

Since is generated by commutators, it is the smallest subgroup of with that property. The abelianization is the largest abelian group onto which admits a surjective homomorphism.

Free groups and presentations

Definition 5.12 (Free group). The free group on generators : its elements are all reduced words where (the empty word is ), with , and .

(Non-reduced words: reduce them by

  • if , combining ;
  • if an exponent is zero, removing .

Repeat until the word is reduced.)

This is the “largest” group with generators; all others are quotients of . If is generated by , define a homomorphism

Definition 5.13 (Finitely presented group). A finitely generated group is said to be finitely presented if the kernel of is the smallest normal subgroup of containing some finite subset (words in the generators), i.e. the subgroup generated by the ’s and their conjugates .

We write (generators | relations); then .

Example 5.14. (the abelianization of !).

Rings and fields

Now we move on to rings and fields on our way to vector spaces (Artin ch. 3 / Axler ch. 1–2). Groups will return later.

Notes not reached in this lecture, covered at the start of {{LINK:L06}}

Definition 5.15 (Ring). A (commutative) ring is a set with two operations such that

  1. is an abelian group with identity ;
  2. is a (commutative) semigroup with identity , namely
    • for all ,
    • for all ,
    • ( for all , if commutative);
  3. distributive law: for all .

Definition 5.16 (Field). A field is a commutative ring such that for all there exists such that , i.e. is an abelian group rather than a semigroup.

Remark.

  • The ring axioms imply for all . (, plus cancellation.)
  • The trivial ring is the only case where . By convention this is not a field.
  • Most rings of interest to us are commutative. (Matrices are the main exception.)
  • In a field, or . This is not necessarily true in a ring.
  • Hence, in a field, we have the usual properties of cancellation (simplification) for both addition and multiplication.

Definition 5.17 (Ring / field homomorphism). A ring/field homomorphism is a map that respects both operations:

  • (we’ve seen this implies , );
  • ;
  • (this doesn’t follow from , even for fields: consider !).

Proposition 5.18. If is a field homomorphism, then is injective.

Proof. If then there exists such that , so , which implies . So (as an additive group homomorphism), hence is injective. ∎

Example 5.19.

  • , are rings.
  • are fields. So is for prime!! This is denoted when viewed as a field.

Why is a field: if in , then its order is (it divides and is ), so . Hence there exists such that . This gives the inverse!

Polynomials and rational functions

Definition 5.20 (Polynomial ring). Given a field , the ring of polynomials in one formal variable is

Remark. is a formal variable, i.e. not an element of anything, though we can evaluate a polynomial at an element of or of any field containing .

So a polynomial a finite tuple of elements of , with componentwise addition [but not componentwise multiplication! ].

Definition 5.21 (Field of rational functions). isn’t a field, but it can be turned into a field by considering fractions (just like the ring the field ): the field of rational functions is

(This generalizes to polynomials and rational functions in any number of variables.)

Power series and Laurent series

Definition 5.22 (Formal power series). The ring of formal power series in is (Add and multiply just like polynomials, term by term. Check that each coefficient in is a finite expression.)

Lemma 5.23. has a multiplicative inverse in iff .

Proof. We want such that . This gives If , clearly there is no solution; if , we can solve inductively: , , … (each step is ✓). ∎

Since every nonzero element of is of the form where is the first nonzero coefficient, to get a field we just need to allow :

Definition 5.24 (Laurent series). The field of Laurent series is

Adjoining roots

Given a field and a polynomial (of degree ), we can evaluate , , and look for roots such that . If there are none in , we can form a field in which has a root.

Example 5.25. : has no roots, but we can form which is a field: .

Example 5.26. , , where is usually called .