Linear algebra over
Recall: every abelian group is also a -module, i.e. it has an operation
Today: linear algebra over and the classification of finitely generated abelian groups (Artin §14.4–14.7).
Theorem 21.1. Any finitely generated abelian group is isomorphic to a product of cyclic groups
(And using iff , we can rearrange the finite factors, e.g. to arrange that all are powers of primes.)
The strategy for the classification is as follows.
Proposition 21.2 (Prop. 1). If is a finitely generated -module, then there exist and such that . (Equivalently: there is an exact sequence .)
(The quotient by a -submodule is the same as the quotient of an abelian group by a subgroup.)
This relies on:
Lemma 21.3. Any submodule ( subgroup) of is finitely generated (in fact, free of rank ).
Proof. By induction on . True for : the subgroups of are and , .
Assume the result holds for , and consider a submodule . The map
restricts to a homomorphism , where
- is a submodule of , hence finitely generated (free) by induction;
- is a subgroup of , hence free (of rank 0 or 1).
And if and are finitely generated (resp. free), then so is ! The proof is just as in the proof of the rank theorem: let be generators (resp. a basis) of , and generators of . Then for all there exist such that , so , so
hence : the generate. (Basis: left as an exercise; we won’t need it anyway.) ∎
Proof of the proposition. If is finitely generated, with generators , then
is surjective, and is a subgroup/submodule of , hence finitely generated by the lemma. Let be generators of ; then
and now we have an exact sequence , with , inducing an isomorphism . ∎
Divisibility and primitive elements
The next ingredient is the notion of divisibility of an element of (or of a free -module).
Definition 21.4 (Divisibility). The divisibility of a nonzero element is the largest for which there exists such that (i.e. ).
An element of is primitive if its divisibility is .
Lemma 21.5. An element of a free finitely generated -module (e.g. ) can be chosen to be part of a basis iff it is primitive (or: it is times a basis element iff its divisibility is ).
Proof.
- Clearly, elements of a basis are primitive (linear independence prevents for some ).
- Converse: the Euclidean division algorithm. Let be primitive. Without loss of generality assume , . Then let (Euclidean division with remainder), and change basis to
to get , which makes all other coefficients . Repeat this process; in finitely many steps we’re left with times a basis vector.
∎
Proposition 21.6 (Prop. 2). For every , there exist bases of and of , (the rank of ) and positive integers such that
Proof. If the statement is obvious for all . Otherwise, proceed by induction on .
Case : let ; by the lemma there is a basis of such that .
Assume the result is proved for , and consider (we can assume ). Let
and such that . Then is necessarily primitive (if it is divisible by , then ), and write , primitive.
Using the lemma, complete to bases of and of . Now
where is the restriction of to , composed with the projection to .
Use the induction hypothesis: replacing and with some other bases of their span, we can assume
Then
i.e. for some , for (with for ).
Write , and change basis to . Then
Now would give , contradicting our choice of . So for all , and we’re done. ∎
Note. The construction actually gives , because
Proof of the theorem. Prop. 1 implies any finitely generated -module is for some , and Prop. 2 implies that after a change of basis of , we can assume is spanned by for some , . So
∎
Notes not reached in this lecture, covered at the start of {{LINK:L22}}
Group actions
(Artin §6.7.)
Definition 21.7 (Group action). An action of a group on a set is a homomorphism . Equivalently, we have a map
This generalizes the idea of groups as symmetries of geometric objects. Understanding what sets a group acts on (and in what way) gives information about !
Definition 21.8 (Faithful action). An action is faithful if is injective.
(Otherwise, the group that “really” acts on is ...)
Definition 21.9 (Orbit). The orbit of under is .
Observe: such that , and then . So the orbits of the -action form a partition of .
Equivalently: such that is an equivalence relation:
- since ;
- implies there is with ; then , so ;
- and imply there are with and with ; then , hence .
Orbits are the equivalence classes of this relation.
Definition 21.10 (Transitive action). An action is transitive if there is only one orbit, i.e. for all there exists such that .
Note: given any -action on , by restriction we get a -action separately on each orbit. Each of these is transitive (by definition), so we can break up any group action into a disjoint union of transitive actions!
Definition 21.11 (Stabilizers and fixed points).
- The stabilizer of is . This is a subgroup of !
- The fixed points of are the subset .
Proposition 21.12. If then . So: elements in the same orbit have conjugate stabilizers.
Proof. implies , so . Conversely, the same argument for gives , hence equality. ∎
Example 21.13. Given a subgroup , we have a set . To avoid notational confusion, write for elements of .
acts on by left multiplication: . This action is transitive ( maps to ). itself, and .
Claim: this is what a general group action looks like when restricted to an orbit!
Proposition 21.14. If acts on a set , given , let . Then
is a bijection, and equivariant, i.e. it intertwines the -actions: .